1
|
Marick S, Takasu F, Bairagi N. How do productivity gradient and diffusion shape patterns in a plant-herbivore grazing system? J Theor Biol 2024; 590:111856. [PMID: 38777134 DOI: 10.1016/j.jtbi.2024.111856] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2024] [Revised: 05/05/2024] [Accepted: 05/13/2024] [Indexed: 05/25/2024]
Abstract
Natural systems show heterogeneous patchy distributions of vegetation over large landscapes. Reaction-diffusion systems can demonstrate such heterogeneity of species distributions. Here, we analyse a reaction-diffusion model of plant-herbivore interactions in two-dimensional space to illustrate non-homogeneous distributions of plants and herbivores. The non-spatial system shows bottom-up control, where herbivore density is low under low and high primary productivity but increased at intermediate productivity. In addition, the non-spatial system provides bistability between a dense vegetation state devoid of herbivores and a coexisting state of plants and herbivores. In the spatiotemporal model, we give analytical conditions of occurring diffusion-driven (Turing) instability, where a novel point in our model is the relative dispersal of herbivores, which represents the movement of herbivores from a higher to a lower vegetation state in addition to the self-diffusion of both species. It is shown that heterogeneity in the population distribution does not occur if the relative dispersal of herbivores is low, but it appears in the opposite case. Due to bistability in the underlying non-spatial system, the spatiotemporal model produces initial value-dependent patterns. The two initial values make different patterns despite having the same primary productivity and relative dispersal rate. As productivity increases with a given relative herbivore dispersal, pattern transition occurs from a blend of stripes and spots of low vegetation state to a predominantly low-density vegetation state with smaller patches of densely vegetated states with one initial value. On the contrary, a discernible change in vegetation patterns from cold spots in the dense vegetation to hot stripes in the primarily low-vegetated state is noticed under the other initial population value. Furthermore, the population distributions of plants and herbivores in the entire domain after a long period are heterogeneous for both initial values, provided the relative herbivore dispersal is substantial. We estimated mean population densities to observe species fitness in the whole domain under variable productivity. When productivity is high, the mean population density of plants may go up or down, depending on the herbivore's relative dispersal rate. In contrast to the bottom-up control dynamics of the non-spatial system, the system exhibits a top-down control under high relative dispersal, where the herbivore regulates vegetation growth under high productivity. On the other hand, herbivores are extinct under high productivity if the relative dispersal is low.
Collapse
Affiliation(s)
- Sounov Marick
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
| | - Fugo Takasu
- Department of Environmental Science, Nara Women's University, Nara, Japan
| | - Nandadulal Bairagi
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India.
| |
Collapse
|
2
|
Chen Y, Zeng X, Niu B. Spatiotemporal patterns induced by Turing-Hopf interaction and symmetry on a disk. Phys Rev E 2024; 109:024214. [PMID: 38491631 DOI: 10.1103/physreve.109.024214] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2023] [Accepted: 01/23/2024] [Indexed: 03/18/2024]
Abstract
Turing bifurcation and Hopf bifurcation are two important kinds of transitions giving birth to inhomogeneous solutions, in spatial or temporal ways. On a disk, these two bifurcations may lead to equivariant Turing-Hopf bifurcations whose normal forms are given in three different cases in this paper. In addition, we analyzed the possible solutions for each normal form, which can guide us to find solutions with physical significance in real-world systems, and the breathing, standing wave-like, and rotating wave-like patterns are found in a delayed mussel-algae model.
Collapse
Affiliation(s)
- Yaqi Chen
- Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong 264209, People's Republic of China
| | - Xianyi Zeng
- Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015, USA
| | - Ben Niu
- Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong 264209, People's Republic of China
| |
Collapse
|
3
|
Xiao R, Gao Q, Azaele S, Sun Y. Effects of noise on the critical points of Turing instability in complex ecosystems. Phys Rev E 2023; 108:014407. [PMID: 37583214 DOI: 10.1103/physreve.108.014407] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2023] [Accepted: 07/01/2023] [Indexed: 08/17/2023]
Abstract
Noise is ubiquitous in natural and artificial systems. In a noisy environment, the interactions among nodes may fluctuate randomly, leading to more complicated interactions. In this paper we focus on the effects of noise and network topology on the Turing pattern of ecological networks with activator-inhibitor structure, which may be interpreted as prey-predator interactions. Based on the stability theory of stochastic differential equations, a sufficient condition for the uniform state is derived. The analytical results indicate that noise is beneficial for the uniform state. When the ratio between the diffusion coefficients of the predator and prey increases, the ecosystems can exhibit a transition from a uniform stable state to a Turing pattern, while when the ratio decreases, the ecosystems transit from a Turing pattern to a uniform stable state. There are two crucial critical points in Turing patterns, forward and backward. We find that both forward and backward critical points increase as the noise intensity increases. This means that noise favors a stable homogeneous state compared to a state with a heterogeneous pattern, which is consistent with the analytical results. In addition, noise can weaken the hysteresis phenomenon and even eliminate it in some cases. Furthermore, we report that network topology plays an important role in modulating the uniform state of ecosystems, such as the size of prey-predator systems, the network connectivity, and the strength of interaction.
Collapse
Affiliation(s)
- Rui Xiao
- School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
| | - Qingyu Gao
- College of Chemical Engineering, China University of Mining and Technology, Xuzhou 221116, China
| | - Sandro Azaele
- Department of Physics and Astronomy "G. Galileo," University of Padova, Padova Via Francesco Marzolo 8, 35131 Padova, Italy
| | - Yongzheng Sun
- School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
| |
Collapse
|
4
|
Takyi EM, Cooper K, Dreher A, McCrorey C. The (De)Stabilizing effect of juvenile prey cannibalism in a stage-structured model. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:3355-3378. [PMID: 36899585 DOI: 10.3934/mbe.2023158] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
Cannibalism, or intraspecific predation, is the act of an organism consuming another organism of the same species. In predator-prey relationships, there is experimental evidence to support the existence of cannibalism among juvenile prey. In this work, we propose a stage-structured predator-prey system where cannibalism occurs only in the juvenile prey population. We show that cannibalism has both a stabilizing and destabilizing effect depending on the choice of parameters. We perform stability analysis of the system and also show that the system experiences a supercritical Hopf, saddle-node, Bogdanov-Takens and cusp bifurcation. We perform numerical experiments to further support our theoretical findings. We discuss the ecological implications of our results.
Collapse
Affiliation(s)
- Eric M Takyi
- Department of Mathematics and Computer Science, Ursinus College, Collegeville, PA 19426, USA
| | - Kasey Cooper
- Department of Mathematics and Computer Science, Ursinus College, Collegeville, PA 19426, USA
| | - Ava Dreher
- Department of Mathematics and Statistics, Binghamton University, Binghamton, NY 13902, USA
| | - Caroline McCrorey
- Department of Mathematics, Bellarmine University, Louisville, KY 40205, USA
| |
Collapse
|
5
|
Kato Y, Nakao H. Turing instability in quantum activator–inhibitor systems. Sci Rep 2022; 12:15573. [PMID: 36114210 PMCID: PMC9481611 DOI: 10.1038/s41598-022-19010-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2022] [Accepted: 08/23/2022] [Indexed: 11/26/2022] Open
Abstract
Turing instability is a fundamental mechanism of nonequilibrium self-organization. However, despite the universality of its essential mechanism, Turing instability has thus far been investigated mostly in classical systems. In this study, we show that Turing instability can occur in a quantum dissipative system and analyze its quantum features such as entanglement and the effect of measurement. We propose a degenerate parametric oscillator with nonlinear damping in quantum optics as a quantum activator–inhibitor unit and demonstrate that a system of two such units can undergo Turing instability when diffusively coupled with each other. The Turing instability induces nonuniformity and entanglement between the two units and gives rise to a pair of nonuniform states that are mixed due to quantum noise. Further performing continuous measurement on the coupled system reveals the nonuniformity caused by the Turing instability. Our results extend the universality of the Turing mechanism to the quantum realm and may provide a novel perspective on the possibility of quantum nonequilibrium self-organization and its application in quantum technologies.
Collapse
|
6
|
Bashkirtseva I, Pankratov A, Ryashko L. Noise-induced formation of heterogeneous patterns in the Turing stability zones of diffusion systems. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:444001. [PMID: 36001986 DOI: 10.1088/1361-648x/ac8c77] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2022] [Accepted: 08/24/2022] [Indexed: 06/15/2023]
Abstract
We study a phenomenon of stochastic generation of waveform patterns for reaction-diffusion systems in the Turing stability zone where the homogeneous equilibrium is a single attractor. In this analysis, we use a distributed variant of the Selkov glycolytic model with diffusion and random forcing. It is shown that in the Turing stability zone, random disturbances can induce a diversity of metastable spatial patterns with different waveforms. We carry out the parametric analysis of statistical characteristics of evolution of these patterns, and reveal the dominant patterns in the stochastic flow of mixed spatial structures.
Collapse
Affiliation(s)
| | | | - Lev Ryashko
- Ural Federal University, Lenina, 51, 620000 Ekaterinburg, Russia
| |
Collapse
|
7
|
Chang L, Guo L, Liu C, Wang Z, Sun G. The qualitative and quantitative relationships between pattern formation and average degree in networked reaction-diffusion systems. CHAOS (WOODBURY, N.Y.) 2022; 32:093129. [PMID: 36182400 DOI: 10.1063/5.0107504] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2022] [Accepted: 08/26/2022] [Indexed: 06/16/2023]
Abstract
The Turing pattern is an important dynamic behavior characteristic of activator-inhibitor systems. Differentiating from traditional assumption of activator-inhibitor interactions in a spatially continuous domain, a Turing pattern in networked reaction-diffusion systems has received much attention during the past few decades. In spite of its great progress, it still fails to evaluate the precise influences of network topology on pattern formation. To this end, we try to promote the research on this important and interesting issue from the point of view of average degree-a critical topological feature of networks. We first qualitatively analyze the influence of average degree on pattern formation. Then, a quantitative relationship between pattern formation and average degree, the exponential decay of pattern formation, is proposed via nonlinear regression. The finding holds true for several activator-inhibitor systems including biology model, ecology model, and chemistry model. The significance of this study lies that the exponential decay not only quantitatively depicts the influence of average degree on pattern formation, but also provides the possibility for predicting and controlling pattern formation.
Collapse
Affiliation(s)
- Lili Chang
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China
| | - Luyao Guo
- School of Mathematics, Southeast University, Nanjing 210096, China
| | - Chen Liu
- School of Ecology and Environment Science, Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China
| | - Zhen Wang
- Center for Optical Imagery Analysis and Learning, Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China
| | - Guiquan Sun
- Department of Mathematics, North University of China, Taiyuan 030051, Shanxi, China
| |
Collapse
|
8
|
Boundary Conditions Cause Different Generic Bifurcation Structures in Turing Systems. Bull Math Biol 2022; 84:101. [PMID: 35953624 PMCID: PMC9372019 DOI: 10.1007/s11538-022-01055-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2022] [Accepted: 07/15/2022] [Indexed: 11/13/2022]
Abstract
Turing’s theory of morphogenesis is a generic mechanism to produce spatial patterning from near homogeneity. Although widely studied, we are still able to generate new results by returning to common dogmas. One such widely reported belief is that the Turing bifurcation occurs through a pitchfork bifurcation, which is true under zero-flux boundary conditions. However, under fixed boundary conditions, the Turing bifurcation becomes generically transcritical. We derive these algebraic results through weakly nonlinear analysis and apply them to the Schnakenberg kinetics. We observe that the combination of kinetics and boundary conditions produce their own uncommon boundary complexities that we explore numerically. Overall, this work demonstrates that it is not enough to only consider parameter perturbations in a sensitivity analysis of a specific application. Variations in boundary conditions should also be considered.
Collapse
|
9
|
Massing JC, Gross T. Generalized Structural Kinetic Modeling: A Survey and Guide. Front Mol Biosci 2022; 9:825052. [PMID: 35573734 PMCID: PMC9098827 DOI: 10.3389/fmolb.2022.825052] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2021] [Accepted: 03/29/2022] [Indexed: 11/13/2022] Open
Abstract
Many current challenges involve understanding the complex dynamical interplay between the constituents of systems. Typically, the number of such constituents is high, but only limited data sources on them are available. Conventional dynamical models of complex systems are rarely mathematically tractable and their numerical exploration suffers both from computational and data limitations. Here we review generalized modeling, an alternative approach for formulating dynamical models to gain insights into dynamics and bifurcations of uncertain systems. We argue that this approach deals elegantly with the uncertainties that exist in real world data and enables analytical insight or highly efficient numerical investigation. We provide a survey of recent successes of generalized modeling and a guide to the application of this modeling approach in future studies such as complex integrative ecological models.
Collapse
Affiliation(s)
- Jana C. Massing
- Helmholtz Institute for Functional Marine Biodiversity at the University of Oldenburg (HIFMB), Oldenburg, Germany
- Helmholtz Centre for Marine and Polar Research, Alfred-Wegener-Institute, Bremerhaven, Germany
- Institute for Chemistry and Biology of the Marine Environment (ICBM), Carl-von-Ossietzky University, Oldenburg, Germany
- *Correspondence: Jana C. Massing,
| | - Thilo Gross
- Helmholtz Institute for Functional Marine Biodiversity at the University of Oldenburg (HIFMB), Oldenburg, Germany
- Helmholtz Centre for Marine and Polar Research, Alfred-Wegener-Institute, Bremerhaven, Germany
- Institute for Chemistry and Biology of the Marine Environment (ICBM), Carl-von-Ossietzky University, Oldenburg, Germany
| |
Collapse
|
10
|
Barter E, Brechtel A, Drossel B, Gross T. A closed form for Jacobian reconstruction from time series and its application as an early warning signal in network dynamics. Proc Math Phys Eng Sci 2022; 477:20200742. [PMID: 35153548 PMCID: PMC8300673 DOI: 10.1098/rspa.2020.0742] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2020] [Accepted: 02/18/2021] [Indexed: 11/12/2022] Open
Abstract
The Jacobian matrix of a dynamical system describes its response to perturbations. Conversely, one can estimate the Jacobian matrix by carefully monitoring how the system responds to environmental noise. We present a closed-form analytical solution for the calculation of a system’s Jacobian from a time series. Being able to access the Jacobian enables a broad range of mathematical analyses by which deeper insights into the system can be gained. Here we consider in particular the computation of the leading Jacobian eigenvalue as an early warning signal for critical transitions. To illustrate this approach, we apply it to ecological meta-foodweb models, which are strongly nonlinear dynamical multi-layer networks. Our analysis shows that accurate results can be obtained, although the data demand of the method is still high.
Collapse
Affiliation(s)
- Edmund Barter
- Department of Engineering Mathematics, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, UK
| | - Andreas Brechtel
- TU Darmstadt, Fachbereich Physik, Hochschulstrasse, 6, 64289 Darmstadt, Germany
| | - Barbara Drossel
- TU Darmstadt, Fachbereich Physik, Hochschulstrasse, 6, 64289 Darmstadt, Germany
| | - Thilo Gross
- HIFMB Helmholtz Institute for Functional Marine Biodiversity, Ammerländer Heerstr. 231, Oldenburg, Germany.,Alfred-Wegener-Institute for Marine and Polar Research, Am Handelshaven 12, Bremerhaven, Germany.,University of Oldenburg, Institute for Chemistry and Biology of the Marine Environment, Carl-von-Ossietzky 9-11 Str., Germany.,UC Davis, Department of Computer Science, 1 Shields Avenue Davis, CA 95616, USA
| |
Collapse
|
11
|
Yang GX, Li XY. Effect of delay on pattern formation of a Rosenzweig–MacArthur type reaction–diffusion model with spatiotemporal delay. INT J BIOMATH 2021. [DOI: 10.1142/s1793524522500085] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, a predator–prey reaction–diffusion model with Rosenzweig–MacArthur type functional response and spatiotemporal delay is investigated through using the tool of Turing bifurcation theories. First, by taking the average time delay as a bifurcation parameter, conditions of occurrence of Turing bifurcation are obtained through employing the Routh–Hurwitz criteria. Second, as the average time delay varies the amplitude equations of Turing bifurcation patterns including spots pattern and stripes pattern are also obtained through the multiple scale perturbation method. Finally, the two kinds of spatiotemporal evolution distributions of species such as spots pattern and stripes pattern are shown to illustrate theoretical results.
Collapse
Affiliation(s)
- Gao-Xiang Yang
- School of Mathematics and Statistics, Ankang University, Ankang, Shaanxi 725000, P. R. China
- Institute of Mathematics and Applied Mathematics, Ankang University, Ankang, Shaanxi 725000, P. R. China
| | - Xiao-Yu Li
- School of Computer Science and Engineering, Xi’an University of Technology, Xi’an, Shaanxi 710048, P. R. China
| |
Collapse
|
12
|
Abstract
Early warning signals (EWS) of tipping points are vital to anticipate system collapse or other sudden shifts. However, existing generic early warning indicators designed to work across all systems do not provide information on the state that lies beyond the tipping point. Our results show how deep learning algorithms (artificial intelligence) can provide EWS of tipping points in real-world systems. The algorithm predicts certain qualitative aspects of the new state, and is also more sensitive and generates fewer false positives than generic indicators. We use theory about system behavior near tipping points so that the algorithm does not require data from the study system but instead learns from a universe of possible models. Many natural systems exhibit tipping points where slowly changing environmental conditions spark a sudden shift to a new and sometimes very different state. As the tipping point is approached, the dynamics of complex and varied systems simplify down to a limited number of possible “normal forms” that determine qualitative aspects of the new state that lies beyond the tipping point, such as whether it will oscillate or be stable. In several of those forms, indicators like increasing lag-1 autocorrelation and variance provide generic early warning signals (EWS) of the tipping point by detecting how dynamics slow down near the transition. But they do not predict the nature of the new state. Here we develop a deep learning algorithm that provides EWS in systems it was not explicitly trained on, by exploiting information about normal forms and scaling behavior of dynamics near tipping points that are common to many dynamical systems. The algorithm provides EWS in 268 empirical and model time series from ecology, thermoacoustics, climatology, and epidemiology with much greater sensitivity and specificity than generic EWS. It can also predict the normal form that characterizes the oncoming tipping point, thus providing qualitative information on certain aspects of the new state. Such approaches can help humans better prepare for, or avoid, undesirable state transitions. The algorithm also illustrates how a universe of possible models can be mined to recognize naturally occurring tipping points.
Collapse
|
13
|
Chowdhury PR, Petrovskii S, Banerjee M. Oscillations and Pattern Formation in a Slow-Fast Prey-Predator System. Bull Math Biol 2021; 83:110. [PMID: 34535836 DOI: 10.1007/s11538-021-00941-0] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2021] [Accepted: 08/27/2021] [Indexed: 10/20/2022]
Abstract
We consider the properties of a slow-fast prey-predator system in time and space. We first argue that the simplicity of the prey-predator system is apparent rather than real and there are still many of its hidden properties that have been poorly studied or overlooked altogether. We further focus on the case where, in the slow-fast system, the prey growth is affected by a weak Allee effect. We first consider this system in the non-spatial case and make its comprehensive study using a variety of mathematical techniques. In particular, we show that the interplay between the Allee effect and the existence of multiple timescales may lead to a regime shift where small-amplitude oscillations in the population abundances abruptly change to large-amplitude oscillations. We then consider the spatially explicit slow-fast prey-predator system and reveal the effect of different timescales on the pattern formation. We show that a decrease in the timescale ratio may lead to another regime shift where the spatiotemporal pattern becomes spatially correlated, leading to large-amplitude oscillations in spatially average population densities and potential species extinction.
Collapse
Affiliation(s)
- Pranali Roy Chowdhury
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, 208016, India
| | - Sergei Petrovskii
- School of Computing and Mathematical Sciences, University of Leicester, Leicester, LE1 7RH, UK.,Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, Russian Federation, 117198
| | - Malay Banerjee
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, 208016, India.
| |
Collapse
|
14
|
Kundu S, Muruganandam P, Ghosh D, Lakshmanan M. Amplitude-mediated spiral chimera pattern in a nonlinear reaction-diffusion system. Phys Rev E 2021; 103:062209. [PMID: 34271612 DOI: 10.1103/physreve.103.062209] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2021] [Accepted: 05/21/2021] [Indexed: 11/07/2022]
Abstract
Formation of diverse patterns in spatially extended reaction-diffusion systems is an important aspect of study that is pertinent to many chemical and biological processes. Of special interest is the peculiar phenomenon of chimera state having spatial coexistence of coherent and incoherent dynamics in a system of identically interacting individuals. In the present article, we report the emergence of various collective dynamical patterns while considering a system of prey-predator dynamics in the presence of a two-dimensional diffusive environment. Particularly, we explore the observance of four distinct categories of spatial arrangements among the species, namely, spiral wave, spiral chimera, completely synchronized oscillations, and oscillation death states in a broad region of the diffusion-driven parameter space. Emergence of amplitude-mediated spiral chimera states displaying drifted amplitudes and phases in the incoherent subpopulation is detected for parameter values beyond both Turing and Hopf bifurcations. Transition scenarios among all these distinguishable patterns are numerically demonstrated for a wide range of the diffusion coefficients which reveal that the chimera states arise during the transition from oscillatory to steady-state dynamics. Furthermore, we characterize the occurrence of each of the recognizable patterns by estimating the strength of incoherent subpopulations in the two-dimensional space.
Collapse
Affiliation(s)
- Srilena Kundu
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata-700108, India
| | | | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata-700108, India
| | - M Lakshmanan
- Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli-620024, India
| |
Collapse
|
15
|
Rana S, Bhowmick AR, Sardar T. Invasive dynamics for a predator-prey system with Allee effect in both populations and a special emphasis on predator mortality. CHAOS (WOODBURY, N.Y.) 2021; 31:033150. [PMID: 33810739 DOI: 10.1063/5.0035566] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2020] [Accepted: 03/09/2021] [Indexed: 06/12/2023]
Abstract
We considered a non-linear predator-prey model with an Allee effect on both populations on a two spatial dimension reaction-diffusion setup. Special importance to predator mortality was given as it may be often controlled through human-made harvesting processes. The local dynamics of the model was studied through boundedness, equilibrium, and stability analysis. An extensive numerical stability analysis was performed and found that bi-stability is not possible for the non-spatial model. By analyzing the spatial model, we found the condition for successful invasion and the persistence region of the species based on the predator Allee effect and its mortality parameter. Four different dynamics in this region of the parameter space are mainly explored. First, the Allee effect on both populations leads to various new types of species spread. Second, for a high value of per-capita growth rate, two completely new spreads (e.g., sun surface, colonial) have been found depending on the Allee effect parameter. Third, the Allee coefficient on the predator population leads to spatiotemporal chaos via a patchy spread for both linear and quadratic mortality rates. Finally, a more rigorous analysis is performed to study the chaotic nature of the system within the whole persistence domain. We have studied the possibility of chaos through temporal variation in different invasion regions. Furthermore, the chaotic fluctuation is studied through the sensitivity of initial conditions and by investigating the dominant Lyapunov exponent value.
Collapse
Affiliation(s)
- Sourav Rana
- Department of Statistics, Visva-Bharati University, Santiniketan 731235, India
| | | | - Tridip Sardar
- Department of Mathematics, Dinabandhu Andrews College, Kolkata 700084, India
| |
Collapse
|
16
|
Duran-Nebreda S, Pla J, Vidiella B, Piñero J, Conde-Pueyo N, Solé R. Synthetic Lateral Inhibition in Periodic Pattern Forming Microbial Colonies. ACS Synth Biol 2021; 10:277-285. [PMID: 33449631 PMCID: PMC8486170 DOI: 10.1021/acssynbio.0c00318] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Multicellular entities are characterized by intricate spatial patterns, intimately related to the functions they perform. These patterns are often created from isotropic embryonic structures, without external information cues guiding the symmetry breaking process. Mature biological structures also display characteristic scales with repeating distributions of signals or chemical species across space. Many candidate patterning modules have been used to explain processes during development and typically include a set of interacting and diffusing chemicals or agents known as morphogens. Great effort has been put forward to better understand the conditions in which pattern-forming processes can occur in the biological domain. However, evidence and practical knowledge allowing us to engineer symmetry-breaking is still lacking. Here we follow a different approach by designing a synthetic gene circuit in E. coli that implements a local activation long-range inhibition mechanism. The synthetic gene network implements an artificial differentiation process that changes the physicochemical properties of the agents. Using both experimental results and modeling, we show that the proposed system is capable of symmetry-breaking leading to regular spatial patterns during colony growth. Studying how these patterns emerge is fundamental to further our understanding of the evolution of biocomplexity and the role played by self-organization. The artificial system studied here and the engineering perspective on embryogenic processes can help validate developmental theories and identify universal properties underpinning biological pattern formation, with special interest for the area of synthetic developmental biology.
Collapse
Affiliation(s)
- Salva Duran-Nebreda
- Institut de Biologia Evolutiva (CSIC-UPF), 08003 Barcelona, Spain
- ICREA-Complex Systems Lab, Universitat Pompeu Fabra, 08003 Barcelona, Spain
- Evolution of Technology Lab, Institut de Biologia Evolutiva (CSIC-UPF), 08003 Barcelona, Spain
| | - Jordi Pla
- Institut de Biologia Evolutiva (CSIC-UPF), 08003 Barcelona, Spain
- ICREA-Complex Systems Lab, Universitat Pompeu Fabra, 08003 Barcelona, Spain
| | - Blai Vidiella
- Institut de Biologia Evolutiva (CSIC-UPF), 08003 Barcelona, Spain
- ICREA-Complex Systems Lab, Universitat Pompeu Fabra, 08003 Barcelona, Spain
| | - Jordi Piñero
- Institut de Biologia Evolutiva (CSIC-UPF), 08003 Barcelona, Spain
- ICREA-Complex Systems Lab, Universitat Pompeu Fabra, 08003 Barcelona, Spain
| | - Nuria Conde-Pueyo
- Institut de Biologia Evolutiva (CSIC-UPF), 08003 Barcelona, Spain
- ICREA-Complex Systems Lab, Universitat Pompeu Fabra, 08003 Barcelona, Spain
| | - Ricard Solé
- Institut de Biologia Evolutiva (CSIC-UPF), 08003 Barcelona, Spain
- ICREA-Complex Systems Lab, Universitat Pompeu Fabra, 08003 Barcelona, Spain
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, United States
| |
Collapse
|
17
|
Djouda BS, Ndjomatchoua FT, Moukam Kakmeni FM, Tchawoua C, Tonnang HEZ. Understanding biological control with entomopathogenic fungi-Insights from a stochastic pest-pathogen model. CHAOS (WOODBURY, N.Y.) 2021; 31:023126. [PMID: 33653067 DOI: 10.1063/5.0019971] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2020] [Accepted: 01/27/2021] [Indexed: 06/12/2023]
Abstract
In this study, an individual-based model is proposed to investigate the effect of demographic stochasticity on biological control using entomopathogenic fungi. The model is formulated as a continuous time Markov process, which is then decomposed into a deterministic dynamics using stochastic corrections and system size expansion. The stability and bifurcation analysis shows that the system dynamic is strongly affected by the contagion rate and the basic reproduction number. However, sensitivity analysis of the extinction probability shows that the persistence of a biological control agent depends to the proportion of spores collected from insect cadavers as well as their ability to be reactivated and infect insects. When considering the migration of each species within a set of patches, the dispersion relation shows a Hopf-damped Turing mode for a threshold contagion rate. A large size population led to a spatial and temporal resonant stochasticity and also induces an amplification effect on power spectrum density.
Collapse
Affiliation(s)
- Byliole S Djouda
- Laboratory of Mechanics, Materials and Structures, Research and Postgraduate Training Unit for Physics and Applications, Postgraduate School of Science, Technology and Geosciences, Department of Physics, Faculty of Science, University of Yaoundé 1, P.O. Box 812, Ngoa Ekelle, Yaoundé, Cameroon
| | - Frank T Ndjomatchoua
- Sustainable Impact Platform, Adaptive Agronomy and Pest Ecology Cluster, International Rice Research Institute (IRRI), DAPO Box 7777-1301, Metro Manila, Philippines
| | - F M Moukam Kakmeni
- Complex Systems and Theoretical Biology Group, Laboratory of Research on Advanced Materials and Nonlinear Science (LaRAMaNS), Department of Physics, Faculty of Science, University of Buéa, P.O. Box 63, Buéa, Cameroon
| | - Clément Tchawoua
- Laboratory of Mechanics, Materials and Structures, Research and Postgraduate Training Unit for Physics and Applications, Postgraduate School of Science, Technology and Geosciences, Department of Physics, Faculty of Science, University of Yaoundé 1, P.O. Box 812, Ngoa Ekelle, Yaoundé, Cameroon
| | - Henri E Z Tonnang
- International Institute of Tropical Agriculture (IITA), 08 BP 0932 Tri Postal Abomey Calavi, Cotonou, Benin
| |
Collapse
|
18
|
Gross T, Allhoff KT, Blasius B, Brose U, Drossel B, Fahimipour AK, Guill C, Yeakel JD, Zeng F. Modern models of trophic meta-communities. Philos Trans R Soc Lond B Biol Sci 2020; 375:20190455. [PMID: 33131442 PMCID: PMC7662193 DOI: 10.1098/rstb.2019.0455] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 03/17/2020] [Indexed: 02/06/2023] Open
Abstract
Dispersal and foodweb dynamics have long been studied in separate models. However, over the past decades, it has become abundantly clear that there are intricate interactions between local dynamics and spatial patterns. Trophic meta-communities, i.e. meta-foodwebs, are very complex systems that exhibit complex and often counterintuitive dynamics. Over the past decade, a broad range of modelling approaches have been used to study these systems. In this paper, we review these approaches and the insights that they have revealed. We focus particularly on recent papers that study trophic interactions in spatially extensive settings and highlight the common themes that emerged in different models. There is overwhelming evidence that dispersal (and particularly intermediate levels of dispersal) benefits the maintenance of biodiversity in several different ways. Moreover, some insights have been gained into the effect of different habitat topologies, but these results also show that the exact relationships are much more complex than previously thought, highlighting the need for further research in this area. This article is part of the theme issue 'Integrative research perspectives on marine conservation'.
Collapse
Affiliation(s)
- Thilo Gross
- University of California Davis, Department of Computer Science, 1 Shields Avenue, Davis, CA 95616, USA
- Alfred Wegener Institut. Helmholtz Zentrum für Polar und Meeresforschung, Am Handelshafen 12, 27570 Bremerhaven, Germany
- Univeristät Oldenburg, Institut für Chemie und Biologie des Meeres, Carl-von-Ossietzky-Strasse 9-11, 26111 Oldenburg, Germany
- Helmholtz Institute for Functional Marine Bidiversity, Ammerländer Heerstrasse 231, Oldenburg, Germany
| | - Korinna T. Allhoff
- Universität Tübingen, Department of Biology, Auf der Morgenstelle 5, 72076 Tübingen, Germany
| | - Bernd Blasius
- Alfred Wegener Institut. Helmholtz Zentrum für Polar und Meeresforschung, Am Handelshafen 12, 27570 Bremerhaven, Germany
- Univeristät Oldenburg, Institut für Chemie und Biologie des Meeres, Carl-von-Ossietzky-Strasse 9-11, 26111 Oldenburg, Germany
| | - Ulrich Brose
- German Centre for Integrative Biodiversity Research, Deutscher Platz 5e, 04103 Leipzig, Germany
- Institute for Biodiversity, Friedrich Schiller University Jena, Dornburger-Strasse 159, 07743 Jena, Germany
| | - Barbara Drossel
- TU Darmstadt, Institut für Festkörperphysik, Hochschulstrasse 6, 64289 Darmstadt, Germany
| | - Ashkaan K. Fahimipour
- National Oceanic and Atmospheric Administration, Southwest Fisheries Science Center, 110 McAllister Way, Santa Cruz, CA 95060, USA
| | - Christian Guill
- Universität Potsdam, Institut für Biochemie und Biologie, Karl-Liebknecht-Strasse 24-25, 14476 Potsdam, Germany
| | - Justin D. Yeakel
- University of California, Merced, School of Natural Sciences, 5200 North Lake Road, Merced, CA 95343, USA
| | - Fanqi Zeng
- University of Bristol, Department of Engineering Mathematics, Merchant Venturers Building, Bristol BS8 1UB, UK
| |
Collapse
|
19
|
LI YAN, ZHANG LINYAN, LI DAGEN, SHI HONGBO. SPATIOTEMPORAL DYNAMICS OF A DIFFUSIVE LESLIE-TYPE PREDATOR–PREY MODEL WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE. J BIOL SYST 2020. [DOI: 10.1142/s0218339020500175] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we study the spatiotemporal dynamics of a diffusive Leslie-type predator–prey system with Beddington–DeAngelis functional response under homogeneous Neumann boundary conditions. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the diffusive model, firstly, it is shown that Turing (diffusion-driven) instability occurs which induces spatial inhomogeneous patterns. Next, it is proved that the diffusive model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the diffusive model undergoes Turing–Hopf bifurcation and exhibits spatiotemporal patterns. Numerical simulations are also presented to verify the theoretical results.
Collapse
Affiliation(s)
- YAN LI
- School of Mathematics and Statistics, Huaiyin Normal University, Huaian, Jiangsu 223300, P. R. China
| | - LINYAN ZHANG
- School of Mathematics and Statistics, Huaiyin Normal University, Huaian, Jiangsu 223300, P. R. China
| | - DAGEN LI
- School of Mathematics and Statistics, Huaiyin Normal University, Huaian, Jiangsu 223300, P. R. China
| | - HONG-BO SHI
- School of Mathematics and Statistics, Huaiyin Normal University, Huaian, Jiangsu 223300, P. R. China
| |
Collapse
|
20
|
Taylor NP, Kim H, Krause AL, Van Gorder RA. A Non-local Cross-Diffusion Model of Population Dynamics I: Emergent Spatial and Spatiotemporal Patterns. Bull Math Biol 2020; 82:112. [PMID: 32780350 DOI: 10.1007/s11538-020-00786-z] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2019] [Accepted: 07/31/2020] [Indexed: 11/28/2022]
Abstract
We extend a spatially non-local cross-diffusion model of aggregation between multiple species with directed motion toward resource gradients to include many species and more general kinds of dispersal. We first consider diffusive instabilities, determining that for directed motion along fecundity gradients, the model permits the Turing instability leading to colony formation and persistence provided there are three or more interacting species. We also prove that such patterning is not possible in the model under the Turing mechanism for two species under directed motion along fecundity gradients, confirming earlier findings in the literature. However, when the directed motion is not along fecundity gradients, for instance, if foraging or migration is sub-optimal relative to fecundity gradients, we find that very different colony structures can emerge. This generalization also permits colony formation for two interacting species. In the advection-dominated case, aggregation patterns are more broad and global in nature, due to the inherent non-local nature of the advection which permits directed motion over greater distances, whereas in the diffusion-dominated case, more highly localized patterns and colonies develop, owing to the localized nature of random diffusion. We also consider the interplay between Turing patterning and spatial heterogeneity in resources. We find that for small spatial variations, there will be a combination of Turing patterns and patterning due to spatial forcing from the resources, whereas for large resource variations, spatial or spatiotemporal patterning can be modified greatly from what is predicted on homogeneous domains. For each of these emergent behaviors, we outline the theoretical mechanism leading to colony formation and then provide numerical simulations to illustrate the results. We also discuss implications this model has for studies of directed motion in different ecological settings.
Collapse
Affiliation(s)
- Nick P Taylor
- Department of Plant Sciences, University of Cambridge, Downing Street, Cambridge, CB2 3EA, UK
| | - Hyunyeon Kim
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Andrew L Krause
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Robert A Van Gorder
- Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, 9054, New Zealand.
| |
Collapse
|
21
|
Potts JR, Schlägel UE. Parametrizing diffusion‐taxis equations from animal movement trajectories using step selection analysis. Methods Ecol Evol 2020. [DOI: 10.1111/2041-210x.13406] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Jonathan R. Potts
- School of Mathematics and Statistics University of Sheffield Sheffield UK
| | - Ulrike E. Schlägel
- Plant Ecology and Nature Conservation Institute of Biochemistry and Biology University of Potsdam Potsdam Germany
| |
Collapse
|
22
|
Aldurayhim A, Elsonbaty A, Elsadany AA. Dynamics of diffusive modified Previte-Hoffman food web model. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2020; 17:4225-4256. [PMID: 32987577 DOI: 10.3934/mbe.2020234] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
This paper formulates and analyzes a modified Previte-Hoffman food web with mixed functional responses. We investigate the existence, uniqueness, positivity and boundedness of the proposed model's solutions. The asymptotic local and global stability of the steady states are discussed. Analytical study of the proposed model reveals that it can undergo supercritical Hopf bifurcation. Furthermore, analysis of Turing instability in spatiotemporal version of the model is carried out where regions of pattern creation in parameters space are obtained. Using detailed numerical simulations for the diffusive and non-diffusive cases, the theoretical findings are verified for distinct sets of parameters.
Collapse
Affiliation(s)
- A Aldurayhim
- Mathematics Department, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia
| | - A Elsonbaty
- Mathematics Department, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia
- Department of Engineering Mathematics and Physics, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt
| | - A A Elsadany
- Mathematics Department, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia
- Department of Basic Science, Faculty of Computers and Informatics, Suez Canal University, Ismailia 41522, Egypt
| |
Collapse
|
23
|
Van Gorder RA. Turing and Benjamin–Feir instability mechanisms in non-autonomous systems. Proc Math Phys Eng Sci 2020. [DOI: 10.1098/rspa.2020.0003] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
The Turing and Benjamin–Feir instabilities are two of the primary instability mechanisms useful for studying the transition from homogeneous states to heterogeneous spatial or spatio-temporal states in reaction–diffusion systems. We consider the case when the underlying reaction–diffusion system is non-autonomous or has a base state which varies in time, as in this case standard approaches, which rely on temporal eigenvalues, break down. We are able to establish respective criteria for the onset of each instability using comparison principles, obtaining inequalities which involve the in general time-dependent model parameters and their time derivatives. In the autonomous limit where the base state is constant in time, our results exactly recover the respective Turing and Benjamin–Feir conditions known in the literature. Our results make the Turing and Benjamin–Feir analysis amenable for a wide collection of applications, and allow one to better understand instabilities emergent due to a variety of non-autonomous mechanisms, including time-varying diffusion coefficients, time-varying reaction rates, time-dependent transitions between reaction kinetics and base states which change in time (such as heteroclinic connections between unique steady states, or limit cycles), to name a few examples.
Collapse
Affiliation(s)
- Robert A. Van Gorder
- Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand
| |
Collapse
|
24
|
Kolinichenko AP, Pisarchik AN, Ryashko LB. Stochastic phenomena in pattern formation for distributed nonlinear systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2020; 378:20190252. [PMID: 32279629 PMCID: PMC7202762 DOI: 10.1098/rsta.2019.0252] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
We study a stochastic spatially extended population model with diffusion, where we find the coexistence of multiple non-homogeneous spatial structures in the areas of Turing instability. Transient processes of pattern generation are studied in detail. We also investigate the influence of random perturbations on the pattern formation. Scenarios of noise-induced pattern generation and stochastic transformations are studied using numerical simulations and modality analysis. This article is part of the theme issue 'Patterns in soft and biological matters'.
Collapse
Affiliation(s)
| | - A. N. Pisarchik
- Centre for Biomedical Technology, Technical University of Madrid, Madrid, Spain
- e-mail:
| | | |
Collapse
|
25
|
Bashkirtseva I, Pankratov A, Slepukhina E, Tsvetkov I. Constructive role of noise and diffusion in an excitable slow-fast population system. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2020; 378:20190253. [PMID: 32279634 PMCID: PMC7202758 DOI: 10.1098/rsta.2019.0253] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
We study the effects of noise and diffusion in an excitable slow-fast population system of the Leslie-Gower type. The phenomenon of noise-induced excitement is investigated in the zone of stable equilibria near the Andronov-Hopf bifurcation with the Canard explosion. The stochastic generation of mixed-mode oscillations is studied by numerical simulation and stochastic sensitivity analysis. Effects of the diffusion are considered for the spatially distributed variant of this slow-fast population model. The phenomenon of the diffusion-induced generation of spatial patterns-attractors in the Turing instability zone is demonstrated. The multistability and variety of transient processes of the pattern formation are discussed. This article is part of the theme issue 'Patterns in soft and biological matters'.
Collapse
Affiliation(s)
| | - A. Pankratov
- Ural Federal University, Ekaterinburg, Russian Federation
| | | | - I. Tsvetkov
- Ural Federal University, Ekaterinburg, Russian Federation
| |
Collapse
|
26
|
Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species. MATHEMATICS 2020. [DOI: 10.3390/math8010101] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
Investigation of interacting populations is an active area of research, and various modeling approaches have been adopted to describe their dynamics. Mathematical models of such interactions using differential equations are capable to mimic the stationary and oscillating (regular or irregular) population distributions. Recently, some researchers have paid their attention to explain the consequences of transient dynamics of population density (especially the long transients) and able to capture such behaviors with simple models. Existence of multiple stationary patches and settlement to a stable distribution after a long quasi-stable transient dynamics can be explained by spatiotemporal models with nonlocal interaction terms. However, the studies of such interesting phenomena for three interacting species are not abundant in literature. Motivated by these facts here we have considered a three species prey–predator model where the predator is generalist in nature as it survives on two prey species. Nonlocalities are introduced in the intra-specific competition terms for the two prey species in order to model the accessibility of nearby resources. Using linear analysis, we have derived the Turing instability conditions for both the spatiotemporal models with and without nonlocal interactions. Validation of such conditions indicates the possibility of existence of stationary spatially heterogeneous distributions for all the three species. Existence of long transient dynamics has been presented under certain parametric domain. Exhaustive numerical simulations reveal various scenarios of stabilization of population distribution due to the presence of nonlocal intra-specific competition for the two prey species. Chaotic oscillation exhibited by the temporal model is significantly suppressed when the populations are allowed to move over their habitat and prey species can access the nearby resources.
Collapse
|
27
|
Abstract
Generalist predators have several food sources and do not depend on one prey species to survive. There has been considerable attention paid by modellers to generalist predator-prey interactions in recent years. Erbach and collaborators in 2013 found a complex dynamics with bistability, limit-cycles and bifurcations in a generalist predator-prey system. In this paper we explore the spatio-temporal dynamics of a reaction-diffusion PDE model for the generalist predator-prey dynamics analyzed by Erbach and colleagues. In particular, we study the Turing and Turing-Hopf pattern formation with special attention to the regime of bistability exhibited by the local model. We derive the conditions for Turing instability and find the region of parameters for which Turing and/or Turing-Hopf instability are possible. By means of numerical simulations, we present the main types of patterns observed for parameters in the Turing domain. In the Turing-Hopf range of the parameters, we observed either stable patterns or homogeneous periodic distributions. Our findings reveal that movement can break the effect of hysteresis observed in the local dynamics, what can have important implication in pest management and species conservation.
Collapse
|
28
|
Vandermeer J, Armbrecht I, de la Mora A, Ennis KK, Fitch G, Gonthier DJ, Hajian-Forooshani Z, Hsieh HY, Iverson A, Jackson D, Jha S, Jiménez-Soto E, Lopez-Bautista G, Larsen A, Li K, Liere H, MacDonald A, Marin L, Mathis KA, Monagan I, Morris JR, Ong T, Pardee GL, Rivera-Salinas IS, Vaiyda C, Williams-Guillen K, Yitbarek S, Uno S, Zemenick A, Philpott SM, Perfecto I. The Community Ecology of Herbivore Regulation in an Agroecosystem: Lessons from Complex Systems. Bioscience 2019. [DOI: 10.1093/biosci/biz127] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
AbstractWhether an ecological community is controlled from above or below remains a popular framework that continues generating interesting research questions and takes on especially important meaning in agroecosystems. We describe the regulation from above of three coffee herbivores, a leaf herbivore (the green coffee scale, Coccus viridis), a seed predator (the coffee berry borer, Hypothenemus hampei), and a plant pathogen (the coffee rust disease, caused by Hemelia vastatrix) by various natural enemies, emphasizing the remarkable complexity involved. We emphasize the intersection of this classical question of ecology with the burgeoning field of complex systems, including references to chaos, critical transitions, hysteresis, basin or boundary collision, and spatial self-organization, all aimed at the applied question of pest control in the coffee agroecosystem.
Collapse
Affiliation(s)
- John Vandermeer
- Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor
| | - Inge Armbrecht
- Department of Biology, Universidad del Valle, Cali, Colombia
| | - Aldo de la Mora
- Department of Entomology, University of California, Riverside
| | - Katherine K Ennis
- Environmental Studies Department, University of California, Santa Cruz
| | - Gordon Fitch
- Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor
| | | | | | - Hsun-Yi Hsieh
- Kellogg Biological Station, Michigan State University, South Gull Lake
| | - Aaron Iverson
- Department of Entomology, Cornell University, Ithaca, New York
| | | | - Shalene Jha
- Department of Integrative Biology, University of Texas, Austin
| | | | | | - Ashley Larsen
- Bren School of Environmental Science and Management, University of California, Santa Barbara
| | - Kevin Li
- Department of Crop Sciences, University of Goettingen, Goettingen, Germany
| | - Heidi Liere
- Department of Biology, University of Seattle, Seattle, Washington
| | - Andrew MacDonald
- Earth Research Institute, University of California, Santa Barbara
| | - Linda Marin
- Independent consultant, Chiapas and Pueblo, Mexico
| | | | - Ivan Monagan
- Department of Ecology, Evolution, and Environmental Biology, Columbia University, in New York, New York
| | - Jonathan R Morris
- School of Environment and Sustainability, University of Michigan, Ann Arbor
| | - Theresa Ong
- Environmental Studies Program, Dartmouth College, Hanover, New Hampshire
| | | | | | - Chatura Vaiyda
- Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor
| | | | - Senay Yitbarek
- Department of Integrative Biology, University of California, Berkeley
| | | | | | - Stacy M Philpott
- Environmental Studies Department, University of California, Santa Cruz
| | - Ivette Perfecto
- School of Environment and Sustainability, University of Michigan, Ann Arbor
| |
Collapse
|
29
|
Zhou Z, Van Gorder RA. Turing Instability and Colony Formation in Spatially Extended Rosenzweig-MacArthur Predator-Prey Models with Allochthonous Resources. Bull Math Biol 2019; 81:5009-5053. [PMID: 31595381 DOI: 10.1007/s11538-019-00667-0] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/12/2019] [Accepted: 09/26/2019] [Indexed: 10/25/2022]
Abstract
While it is somewhat well known that spatial PDE extensions of the Rosenzweig-MacArthur predator-prey model do not admit spatial pattern formation through the Turing mechanism, in this paper we demonstrate that the addition of allochthonous resources into the system can result in spatial patterning and colony formation. We study pattern formation, through Turing and Turing-Hopf mechanisms, in two distinct spatial Rosenzweig-MacArthur models generalized to include allochthonous resources. Both models have previously been shown to admit heterogeneous spatial solutions when prey and allochthonous resources are confined to different regions of the domain, with the predator able to move between the regions. However, pattern formation in such cases is not due to the Turing mechanism, but rather due to the spatial separation between the two resources for the predator. On the other hand, for a variety of applications, a predator can forage over a region where more than one food source is present, and this is the case we study in the present paper. We first consider a three PDE model, consisting of equations for each of a predator, a prey, and an allochthonous resource or subsidy, with all three present over the spatial domain. The second model we consider arises in the study of two independent predator-prey systems in which a portion of the prey in the first system becomes an allochthonous resource for the second system; this is referred to as a predator-prey-quarry-resource-scavenger model. We show that there exist parameter regimes for which these systems admit Turing and Turing-Hopf bifurcations, again resulting in spatial or spatiotemporal patterning and hence colony formation. This is interesting from a modeling standpoint, as the standard spatially extended Rosenzweig-MacArthur predator-prey equations do not permit the Turing instability, and hence, the inclusion of allochthonous resources is one route to realizing colony formation under Rosenzweig-MacArthur kinetics. Concerning the ecological application, we find that spatial patterning occurs when the predator is far more mobile than the prey (reflected in the relative difference between their diffusion parameters), with the prey forming colonies and the predators more uniformly dispersed throughout the domain. We discuss how this spatially heterogeneous patterning, particularly of prey populations, may constitute one way in which the paradox of enrichment is resolved in spatial systems by way of introducing allochthonous resource subsidies in conjunction with spatial diffusion of predator and prey populations.
Collapse
Affiliation(s)
- Zhi Zhou
- Department of Engineering Sciences and Applied Mathematics, McCormick School of Engineering and Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208, USA
| | - Robert A Van Gorder
- Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, 9054, New Zealand.
| |
Collapse
|
30
|
Brechtel A, Gross T, Drossel B. Far-ranging generalist top predators enhance the stability of meta-foodwebs. Sci Rep 2019; 9:12268. [PMID: 31439912 PMCID: PMC6706381 DOI: 10.1038/s41598-019-48731-y] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2019] [Accepted: 08/12/2019] [Indexed: 11/08/2022] Open
Abstract
Identifying stabilizing factors in foodwebs is a long standing challenge with wide implications for community ecology and conservation. Here, we investigate the stability of spatially resolved meta-foodwebs with far-ranging super-predators for whom the whole meta-foodwebs appears to be a single habitat. By using a combination of generalized modeling with a master stability function approach, we are able to efficiently explore the asymptotic stability of large classes of realistic many-patch meta-foodwebs. We show that meta-foodwebs with far-ranging top predators are more stable than those with localized top predators. Moreover, adding far-ranging generalist top predators to a system can have a net stabilizing effect. These results highlight the importance of top predator conservation.
Collapse
Affiliation(s)
- Andreas Brechtel
- Technische Universität Darmstadt, Institute for condensed matter physics, Hochschulstr. 6, Darmstadt, 64289, Germany.
| | - Thilo Gross
- UC Davis, Department of Computer Science, 1 Shields Av, Davis, Ca, 95616, USA
| | - Barbara Drossel
- Technische Universität Darmstadt, Institute for condensed matter physics, Hochschulstr. 6, Darmstadt, 64289, Germany
| |
Collapse
|
31
|
Spatio-temporal secondary instabilities near the Turing-Hopf bifurcation. Sci Rep 2019; 9:11287. [PMID: 31375714 PMCID: PMC6677724 DOI: 10.1038/s41598-019-47584-9] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2019] [Accepted: 07/10/2019] [Indexed: 11/08/2022] Open
Abstract
In this work, we provide a framework to understand and quantify the spatiotemporal structures near the codimension-two Turing-Hopf point, resulting from secondary instabilities of Mixed Mode solutions of the Turing-Hopf amplitude equations. These instabilities are responsible for solutions such as (1) patterns which change their effective wavenumber while they oscillate as well as (2) phase instability combined with a spatial pattern. The quantification of these instabilities is based on the solution of the fourth order polynomial for the dispersion relation, which is solved using perturbation techniques. With the proposed methodology, we were able to identify and numerically corroborate that these two kinds of solutions are generalizations of the well known Eckhaus and Benjamin-Feir-Newell instabilities, respectively. Numerical simulations of the coupled system of real and complex Ginzburg-Landau equations are presented in space-time maps, showing quantitative and qualitative agreement with the predicted stability of the solutions. The relation with spatiotemporal intermittency and chaos is also illustrated.
Collapse
|
32
|
Consolo G, Valenti G. Secondary seed dispersal in the Klausmeier model of vegetation for sloped semi-arid environments. Ecol Modell 2019. [DOI: 10.1016/j.ecolmodel.2019.02.009] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
|
33
|
Mimar S, Juane MM, Park J, Muñuzuri AP, Ghoshal G. Turing patterns mediated by network topology in homogeneous active systems. Phys Rev E 2019; 99:062303. [PMID: 31330727 DOI: 10.1103/physreve.99.062303] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2019] [Indexed: 06/10/2023]
Abstract
Mechanisms of pattern formation-of which the Turing instability is an archetype-constitute an important class of dynamical processes occurring in biological, ecological, and chemical systems. Recently, it has been shown that the Turing instability can induce pattern formation in discrete media such as complex networks, opening up the intriguing possibility of exploring it as a generative mechanism in a plethora of socioeconomic contexts. Yet much remains to be understood in terms of the precise connection between network topology and its role in inducing the patterns. Here we present a general mathematical description of a two-species reaction-diffusion process occurring on different flavors of network topology. The dynamical equations are of the predator-prey class that, while traditionally used to model species population, has also been used to model competition between antagonistic features in social contexts. We demonstrate that the Turing instability can be induced in any network topology by tuning the diffusion of the competing species or by altering network connectivity. The extent to which the emergent patterns reflect topological properties is determined by a complex interplay between the diffusion coefficients and the localization properties of the eigenvectors of the graph Laplacian. We find that networks with large degree fluctuations tend to have stable patterns over the space of initial perturbations, whereas patterns in more homogenous networks are purely stochastic.
Collapse
Affiliation(s)
- Sayat Mimar
- Department of Physics & Astronomy, University of Rochester, Rochester, New York 14607, USA
| | - Mariamo Mussa Juane
- Group of Nonlinear Physics, University of Santiago de Compostela, Santiago de Compostela 15782, Spain
| | - Juyong Park
- Graduate School of Culture Technology, Korea Advanced Institute of Science and Technology, Daejon 305-701, Korea
| | - Alberto P Muñuzuri
- Group of Nonlinear Physics, University of Santiago de Compostela, Santiago de Compostela 15782, Spain
| | - Gourab Ghoshal
- Department of Physics & Astronomy, University of Rochester, Rochester, New York 14607, USA
| |
Collapse
|
34
|
Djouda BS, Moukam Kakmeni FM, Guemkam Ghomsi P, Ndjomatchoua FT, Tchawoua C, Tonnang HEZ. Theoretical analysis of spatial nonhomogeneous patterns of entomopathogenic fungi growth on insect pest. CHAOS (WOODBURY, N.Y.) 2019; 29:053134. [PMID: 31154798 DOI: 10.1063/1.5043612] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2018] [Accepted: 04/15/2019] [Indexed: 06/09/2023]
Abstract
This paper presents the study of the dynamics of intrahost (insect pests)-pathogen [entomopathogenic fungi (EPF)] interactions. The interaction between the resources from the insect pest and the mycelia of EPF is represented by the Holling and Powell type II functional responses. Because the EPF's growth is related to the instability of the steady state solution of our system, particular attention is given to the stability analysis of this steady state. Initially, the stability of the steady state is investigated without taking into account diffusion and by considering the behavior of the system around its equilibrium states. In addition, considering small perturbation of the stable singular point due to nonlinear diffusion, the conditions for Turing instability occurrence are deduced. It is observed that the absence of the regeneration feature of insect resources prevents the occurrence of such phenomena. The long time evolution of our system enables us to observe both spot and stripe patterns. Moreover, when the diffusion of mycelia is slightly modulated by a weak periodic perturbation, the Floquet theory and numerical simulations allow us to derive the conditions in which diffusion driven instabilities can occur. The relevance of the obtained results is further discussed in the perspective of biological insect pest control.
Collapse
Affiliation(s)
- Byliole S Djouda
- Laboratory of Mechanics, Materials and Structures, Research and Postgraduate Training Unit for Physics and Applications, Postgraduate School of Science, Technology and Geosciences, Department of Physics, Faculty of Science, University of Yaoundé 1, P.O. Box 812, Ngoa Ekelle, Yaoundé, Cameroon
| | - F M Moukam Kakmeni
- Complex Systems and Theoretical Biology Group, Laboratory of Research on Advanced Materials and Nonlinear Science (LaRAMaNS), Department of Physics, Faculty of Science, University of Buéa, P. O. Box 63, Buéa, Cameroon
| | - P Guemkam Ghomsi
- Laboratory of Mechanics, Materials and Structures, Research and Postgraduate Training Unit for Physics and Applications, Postgraduate School of Science, Technology and Geosciences, Department of Physics, Faculty of Science, University of Yaoundé 1, P.O. Box 812, Ngoa Ekelle, Yaoundé, Cameroon
| | - Frank T Ndjomatchoua
- Sustainable Impact Platform, Adaptive Agronomy and Pest Ecology Cluster, International Rice Research Institute (IRRI), DAPO Box 7777-1301, Metro Manila, Philippines
| | - Clément Tchawoua
- Laboratory of Mechanics, Materials and Structures, Research and Postgraduate Training Unit for Physics and Applications, Postgraduate School of Science, Technology and Geosciences, Department of Physics, Faculty of Science, University of Yaoundé 1, P.O. Box 812, Ngoa Ekelle, Yaoundé, Cameroon
| | - Henri E Z Tonnang
- International Institute of Tropical Agriculture (IITA), 08 BP 0932, Tri Postal Abomey Calavi, Cotonou, Benin
| |
Collapse
|
35
|
Fussell EF, Krause AL, Van Gorder RA. Hybrid approach to modeling spatial dynamics of systems with generalist predators. J Theor Biol 2019; 462:26-47. [PMID: 30385311 DOI: 10.1016/j.jtbi.2018.10.054] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2018] [Revised: 10/06/2018] [Accepted: 10/29/2018] [Indexed: 11/16/2022]
Abstract
We consider hybrid spatial modeling approaches for ecological systems with a generalist predator utilizing a prey and either a second prey or an allochthonous resource. While spatial dispersion of populations is often modeled via stepping-stone (discrete spatial patches) or continuum (one connected spatial domain) formulations, we shall be interested in hybrid approaches which we use to reduce the dimension of certain components of the spatial domain, obtaining either a continuum model of varying spatial dimensions, or a mixed stepping-stone-continuum model. This approach results in models consisting of partial differential equations for some of the species which are coupled via reactive boundary conditions to lower dimensional partial differential equations or ordinary differential equations for the other species. In order to demonstrate the use of this approach, we consider two case studies. In the first case study, we consider a one-predator two-prey interaction between beavers, wolves and white-tailed deer in Voyageurs National Park. In the second case study, we consider predator-prey-allochthonous resource interactions between bears, berries and salmon on Kodiak Island. For each case study, we compare the results from the hybrid modeling approach with corresponding stepping-stone and continuum model results, highlighting benefits and limitations of the method. In some cases, we find that the hybrid modeling approach allows for solutions which are easier to simulate (akin to stepping-stone models) while maintaining seemingly more realistic spatial dynamics (akin to full continuum models).
Collapse
Affiliation(s)
- Elizabeth F Fussell
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Andrew L Krause
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Robert A Van Gorder
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.
| |
Collapse
|
36
|
Laplacian matrices and Turing bifurcations: revisiting Levin 1974 and the consequences of spatial structure and movement for ecological dynamics. THEOR ECOL-NETH 2019. [DOI: 10.1007/s12080-018-0403-2] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/05/2023]
|
37
|
|
38
|
Mukherjee N, Ghorai S, Banerjee M. Effects of density dependent cross-diffusion on the chaotic patterns in a ratio-dependent prey-predator model. ECOLOGICAL COMPLEXITY 2018. [DOI: 10.1016/j.ecocom.2017.11.006] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
|
39
|
Stationary, non-stationary and invasive patterns for a prey-predator system with additive Allee effect in prey growth. ECOLOGICAL COMPLEXITY 2018. [DOI: 10.1016/j.ecocom.2018.09.001] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
|
40
|
Sánchez-Garduño F, Krause AL, Castillo JA, Padilla P. Turing-Hopf patterns on growing domains: The torus and the sphere. J Theor Biol 2018; 481:136-150. [PMID: 30266461 DOI: 10.1016/j.jtbi.2018.09.028] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2018] [Revised: 09/18/2018] [Accepted: 09/24/2018] [Indexed: 10/28/2022]
Abstract
This paper deals with the study of spatial and spatio-temporal patterns in the reaction-diffusion FitzHugh-Nagumo model on growing curved domains. This is carried out on two exemplar cases: a torus and a sphere. We compute bifurcation boundaries for the homogeneous steady state when the homogeneous system is monostable. We exhibit Turing and Turing-Hopf bifurcations, as well as additional patterning outside of these bifurcation regimes due to the multistability of patterned states. We consider static and growing domains, where the growth is slow, isotropic, and exponential in time, allowing for a simple analytical calculation of these bifurcations in terms of model parameters. Numerical simulations allow us to discuss the role played by the growth and the curvature of the domains on the pattern selection on the torus and the sphere. We demonstrate parameter regimes where the linear theory can successfully predict the kind of pattern (homogeneous and heterogeneous oscillations and stationary spatial patterns) but not their detailed nonlinear structure. We also find parameter regimes where the linear theory fails, such as Hopf regimes which give rise to spatial patterning (depending on geometric details), where we suspect that multistability plays a key role in the departure from homogeneity. Finally we also demonstrate effects due to the evolution of nonuniform patterns under growth, suggesting important roles for growth in reaction-diffusion systems beyond modifying instability regimes.
Collapse
Affiliation(s)
- Faustino Sánchez-Garduño
- Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM), Ciudad Universitaria. Circuito Exterior, Ciudad de México, Delegación Coyoacán CP 04510, México.
| | - Andrew L Krause
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Rd, Oxford OX2 6GG, United Kingdom.
| | - Jorge A Castillo
- Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame # 54, Col. Garita, Acapulco, Guerrero CP 39650, México.
| | - Pablo Padilla
- IIMAS-UNAM, Apartado Postal 20-726, Admon. No. 20, Delegación Álvaro Obregón, Ciudad de México CP 01000, México.
| |
Collapse
|
41
|
Flores J. Decreasing fractal dimensions as a strategy for oceanic wildlife conservation: Application to species with large migration patterns. Ecol Modell 2018. [DOI: 10.1016/j.ecolmodel.2018.06.001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023]
|
42
|
Brechtel A, Gramlich P, Ritterskamp D, Drossel B, Gross T. Master stability functions reveal diffusion-driven pattern formation in networks. Phys Rev E 2018; 97:032307. [PMID: 29776185 DOI: 10.1103/physreve.97.032307] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/04/2017] [Indexed: 11/07/2022]
Abstract
We study diffusion-driven pattern formation in networks of networks, a class of multilayer systems, where different layers have the same topology, but different internal dynamics. Agents are assumed to disperse within a layer by undergoing random walks, while they can be created or destroyed by reactions between or within a layer. We show that the stability of homogeneous steady states can be analyzed with a master stability function approach that reveals a deep analogy between pattern formation in networks and pattern formation in continuous space. For illustration, we consider a generalized model of ecological meta-food webs. This fairly complex model describes the dispersal of many different species across a region consisting of a network of individual habitats while subject to realistic, nonlinear predator-prey interactions. In this example, the method reveals the intricate dependence of the dynamics on the spatial structure. The ability of the proposed approach to deal with this fairly complex system highlights it as a promising tool for ecology and other applications.
Collapse
Affiliation(s)
- Andreas Brechtel
- Institute of Condensed Matter Physics, Technische Universität Darmstadt, 64289 Darmstadt, Germany
| | - Philipp Gramlich
- Institute of Condensed Matter Physics, Technische Universität Darmstadt, 64289 Darmstadt, Germany
| | - Daniel Ritterskamp
- Department of Engineering Mathematics, Merchant Venturers School of Engineering, University of Bristol, Woodland Road, Bristol BS8 1UB, United Kingdom
| | - Barbara Drossel
- Institute of Condensed Matter Physics, Technische Universität Darmstadt, 64289 Darmstadt, Germany
| | - Thilo Gross
- Department of Engineering Mathematics, Merchant Venturers School of Engineering, University of Bristol, Woodland Road, Bristol BS8 1UB, United Kingdom
| |
Collapse
|
43
|
Uppal G, Vural DC. Shearing in flow environment promotes evolution of social behavior in microbial populations. eLife 2018; 7:34862. [PMID: 29785930 PMCID: PMC6002248 DOI: 10.7554/elife.34862] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2018] [Accepted: 05/10/2018] [Indexed: 11/23/2022] Open
Abstract
How producers of public goods persist in microbial communities is a major question in evolutionary biology. Cooperation is evolutionarily unstable, since cheating strains can reproduce quicker and take over. Spatial structure has been shown to be a robust mechanism for the evolution of cooperation. Here we study how spatial assortment might emerge from native dynamics and show that fluid flow shear promotes cooperative behavior. Social structures arise naturally from our advection-diffusion-reaction model as self-reproducing Turing patterns. We computationally study the effects of fluid advection on these patterns as a mechanism to enable or enhance social behavior. Our central finding is that flow shear enables and promotes social behavior in microbes by increasing the group fragmentation rate and thereby limiting the spread of cheating strains. Regions of the flow domain with higher shear admit high cooperativity and large population density, whereas low shear regions are devoid of life due to opportunistic mutations. According to the principle of the ‘survival of the fittest’, selfish individuals should be better off compared to peers that cooperate with each other. Indeed, even though a population of organisms benefits from working together, selfish members can exploit the cooperative behavior of others without doing their part. These ‘cheaters’ then use their advantage to reproduce faster and take over the population. Yet, social cooperation is widespread in the natural world, and occurs in creatures as diverse as bacteria and whales. How can it arise and persist then? One idea is that when individuals form distinct groups, the ones with cheaters will perish. Even though a selfish individual will fare better than the rest of its team, overall, cooperating groups will survive more and reproduce faster; ultimately, they will be favored by evolution. This is called group selection. Here, Uppal and Vural examine how the physical properties of the environment can influence the evolution of social interactions between bacteria. To this end, mathematical models are used to simulate how bacteria grow, evolve and drift in a flowing fluid. These are based on equations worked out from the behavior of real-life populations. The results show that flow patterns in a fluid habitat govern the social behavior of bacteria. When different regions of the fluid are moving at different speeds, ‘shear forces’ are created that cause bacterial colonies to distort and occasionally break apart to form two groups. As such, cooperative groups will rapidly form new cooperating colonies, whereas groups with cheaters will reproduce slower or perish. Furthermore, results show that when different areas of the fluid have different shear forces, social cooperation will only prevail in certain places. This makes it possible to use flow patterns to fine tune social evolution so that cooperating bacteria will be confined in a certain region. Outside of this area, these bacteria would be taken over by cheaters and go extinct. Bacteria are both useful and dangerous to humans: for example, certain species can break down pollutants in the water, when others cause deadly infections. These results show it could be possible to control the activity of these microorganisms to our advantage by changing the flow of the fluids in which they live. More broadly, the simulations developed by Uppal and Vural can be applied to a variety of ecosystems where microscopic organisms inhabit fluids, such as plankton flowing in oceanic currents.
Collapse
Affiliation(s)
- Gurdip Uppal
- Department of Physics, University of Notre Dame, Notre Dame, United States
| | - Dervis Can Vural
- Department of Physics, University of Notre Dame, Notre Dame, United States
| |
Collapse
|
44
|
|
45
|
Gambino G, Lombardo MC, Sammartino M. Cross-diffusion-induced subharmonic spatial resonances in a predator-prey system. Phys Rev E 2018; 97:012220. [PMID: 29448421 DOI: 10.1103/physreve.97.012220] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/22/2017] [Indexed: 06/08/2023]
Abstract
In this paper we investigate the complex dynamics originated by a cross-diffusion-induced subharmonic destabilization of the fundamental subcritical Turing mode in a predator-prey reaction-diffusion system. The model we consider consists of a two-species Lotka-Volterra system with linear diffusion and a nonlinear cross-diffusion term in the predator equation. The taxis term in the search strategy of the predator is responsible for the onset of complex dynamics. In fact, our model does not exhibit any Hopf or wave instability, and on the basis of the linear analysis one should only expect stationary patterns; nevertheless, the presence of the nonlinear cross-diffusion term is able to induce a secondary instability: due to a subharmonic spatial resonance, the stationary primary branch bifurcates to an out-of-phase oscillating solution. Noticeably, the strong resonance between the harmonic and the subharmonic is able to generate the oscillating pattern albeit the subharmonic is below criticality. We show that, as the control parameter is varied, the oscillating solution (subT mode) can undergo a sequence of secondary instabilities, generating a transition toward chaotic dynamics. Finally, we investigate the emergence of subT-mode solutions on two-dimensional domains: when the fundamental mode describes a square pattern, subharmonic resonance originates oscillating square patterns. In the case of subcritical Turing hexagon solutions, the internal interactions with a subharmonic mode are able to generate the so-called "twinkling-eyes" pattern.
Collapse
Affiliation(s)
- G Gambino
- Department of Mathematics, University of Palermo, via Archirafi 34, 90123 Palermo, Italy
| | - M C Lombardo
- Department of Mathematics, University of Palermo, via Archirafi 34, 90123 Palermo, Italy
| | - M Sammartino
- DIID, University of Palermo, Viale delle Scienze, Ed. 8, 90128 Palermo, Italy
| |
Collapse
|
46
|
Petit J, Lauwens B, Fanelli D, Carletti T. Theory of Turing Patterns on Time Varying Networks. PHYSICAL REVIEW LETTERS 2017; 119:148301. [PMID: 29053314 DOI: 10.1103/physrevlett.119.148301] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2017] [Indexed: 06/07/2023]
Abstract
The process of pattern formation for a multispecies model anchored on a time varying network is studied. A nonhomogeneous perturbation superposed to an homogeneous stable fixed point can be amplified following the Turing mechanism of instability, solely instigated by the network dynamics. By properly tuning the frequency of the imposed network evolution, one can make the examined system behave as its averaged counterpart, over a finite time window. This is the key observation to derive a closed analytical prediction for the onset of the instability in the time dependent framework. Continuously and piecewise constant periodic time varying networks are analyzed, setting the framework for the proposed approach. The extension to nonperiodic settings is also discussed.
Collapse
Affiliation(s)
- Julien Petit
- naXys, Namur Institute for Complex Systems, University of Namur, B5000 Namur, Belgium
- Department of Mathematics, Royal Military Academy, B1000 Brussels, Belgium
| | - Ben Lauwens
- Department of Mathematics, Royal Military Academy, B1000 Brussels, Belgium
| | - Duccio Fanelli
- Dipartimento di Fisica e Astronomia and CSDC, Università degli Studi di Firenze, 50019 Sesto Fiorentino, Italy
- INFN Sezione di Firenze, 50019 Sesto Fiorentino, Italy
| | - Timoteo Carletti
- naXys, Namur Institute for Complex Systems, University of Namur, B5000 Namur, Belgium
| |
Collapse
|
47
|
Uthamacumaran A. A biophysical approach to cancer dynamics: Quantum chaos and energy turbulence. Biosystems 2017; 156-157:1-22. [PMID: 28377182 DOI: 10.1016/j.biosystems.2017.03.004] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2016] [Accepted: 03/22/2017] [Indexed: 02/06/2023]
Abstract
Cancer is a term used to define a collective set of rapidly evolving cells with immortalized replication, altered epimetabolomes and patterns of longevity. Identifying a common signaling cascade to target all cancers has been a major obstacle in medicine. A quantum dynamic framework has been established to explain mutation theory, biological energy landscapes, cell communication patterns and the cancer interactome under the influence of quantum chaos. Quantum tunneling in mutagenesis, vacuum energy field dynamics, and cytoskeletal networks in tumor morphogenesis have revealed the applicability for description of cancer dynamics, which is discussed with a brief account of endogenous hallucinogens, bioelectromagnetism and water fluctuations. A holistic model of mathematical oncology has been provided to identify key signaling pathways required for the phenotypic reprogramming of cancer through an epigenetic landscape. The paper will also serve as a mathematical guide to understand the cancer interactome by interlinking theoretical and experimental oncology. A multi-dimensional model of quantum evolution by adaptive selection has been established for cancer biology.
Collapse
|
48
|
Abstract
We study the spatiotemporal dynamics in a diffusive predator–prey system with time delay. By investigating the dynamical behavior of the system in the presence of Turing–Hopf bifurcations, we present a classification of the pattern dynamics based on the dispersion relation for the two unstable modes. More specifically, we researched the existence of the Turing pattern when control parameters lie in the Turing space. Particularly, when parameter values are taken in Turing–Hopf domain, we numerically investigate the formation of all the possible patterns, including time-dependent wave pattern, persistent short-term competing dynamics and stationary Turing pattern. Furthermore, the effect of time delay on the formation of spatial pattern has also been analyzed from the aspects of theory and numerical simulation. We speculate that the interaction of spatial and temporal instabilities in the reaction–diffusion system might bring some insight to the finding of patterns in spatial predator–prey models.
Collapse
Affiliation(s)
- Guangping Hu
- School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, P. R. China
| | - Xiaoling Li
- School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, P. R. China
| | - Dongliang Li
- College of Atmospheric Science, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, P. R. China
| |
Collapse
|
49
|
The complex dynamics of a diffusive prey–predator model with an Allee effect in prey. ECOLOGICAL COMPLEXITY 2016. [DOI: 10.1016/j.ecocom.2016.07.001] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
|
50
|
Herrero A, Stavans J, Flores E. The multicellular nature of filamentous heterocyst-forming cyanobacteria. FEMS Microbiol Rev 2016; 40:831-854. [DOI: 10.1093/femsre/fuw029] [Citation(s) in RCA: 159] [Impact Index Per Article: 19.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Revised: 05/10/2016] [Accepted: 07/09/2016] [Indexed: 11/13/2022] Open
|