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Sherratt JA, Liu QX, van de Koppel J. A Comparison of the "Reduced Losses" and "Increased Production" Models for Mussel Bed Dynamics. Bull Math Biol 2021; 83:99. [PMID: 34427781 PMCID: PMC8384834 DOI: 10.1007/s11538-021-00932-1] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2020] [Accepted: 07/20/2021] [Indexed: 11/02/2022]
Abstract
Self-organised regular pattern formation is one of the foremost examples of the development of complexity in ecosystems. Despite the wide array of mechanistic models that have been proposed to understand pattern formation, there is limited general understanding of the feedback processes causing pattern formation in ecosystems, and how these affect ecosystem patterning and functioning. Here we propose a generalised model for pattern formation that integrates two types of within-patch feedback: amplification of growth and reduction of losses. Both of these mechanisms have been proposed as causing pattern formation in mussel beds in intertidal regions, where dense clusters of mussels form, separated by regions of bare sediment. We investigate how a relative change from one feedback to the other affects the stability of uniform steady states and the existence of spatial patterns. We conclude that there are important differences between the patterns generated by the two mechanisms, concerning both biomass distribution in the patterns and the resilience of the ecosystems to disturbances.
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Affiliation(s)
- Jonathan A Sherratt
- Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK.
| | - Quan-Xing Liu
- State Key Laboratory of Estuarine and Coastal Research, School of Ecological and Environmental Sciences, East China Normal University, Shanghai, 200241, People's Republic of China
| | - Johan van de Koppel
- Department of Estuarine and Delta Systems, Royal Netherlands Institute for Sea Research and Utrecht University, PO Box 140, 4400 AC, Yerseke, The Netherlands
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2
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Tiwari SK, Singh R, Thakur NK. Dynamical study of infochemical influences on tropic interaction of diffusive plankton system. SN APPLIED SCIENCES 2021. [DOI: 10.1007/s42452-021-04237-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022] Open
Abstract
AbstractWe propose a model for tropic interaction among the infochemical-producing phytoplankton and non-info chemical-producing phytoplankton and microzooplankton. Volatile information-conveying chemicals (infochemicals) released by phytoplankton play an important role in the food webs of marine ecosystems. Microzooplankton is an ecologically important grazer of phytoplankton for coexistence of a large number of phytoplankton species. Here, we discuss how information transferred by dimethyl sulfide shapes the interaction of phytoplankton. Phytoplankton deterrents may lead to propagation of IPP bloom. The interaction between IPP and microzooplankton follows the Beddington–DeAngelis-type functional response. Analytically, we discuss boundedness, stability and Turing instability of the model system. We perform numerical simulation for temporal (ODE model) as well as a spatial model system. Our numerical investigation shows that microzooplankton grazing refuse of IPP leads to oscillatory dynamics. Increasing diffusion coefficient of microzooplankton shows Turing instability. Time evolution also plays an important role in the stability of system dynamics. The results obtained in this paper are useful to understand the dominance of algal bloom in coastal and estuarine ecosystem.
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3
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Hori Y, Miyazako H. Analysing diffusion and flow-driven instability using semidefinite programming. J R Soc Interface 2019; 16:20180586. [PMID: 30958171 PMCID: PMC6364638 DOI: 10.1098/rsif.2018.0586] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2018] [Accepted: 01/07/2019] [Indexed: 11/12/2022] Open
Abstract
Diffusion and flow-driven instability, or transport-driven instability, is one of the central mechanisms to generate inhomogeneous gradient of concentrations in spatially distributed chemical systems. However, verifying the transport-driven instability of reaction-diffusion-advection systems requires checking the Jacobian eigenvalues of infinitely many Fourier modes, which is computationally intractable. To overcome this limitation, this paper proposes mathematical optimization algorithms that determine the stability/instability of reaction-diffusion-advection systems by finite steps of algebraic calculations. Specifically, the stability/instability analysis of Fourier modes is formulated as a sum-of-squares optimization program, which is a class of convex optimization whose solvers are widely available as software packages. The optimization program is further extended for facile computation of the destabilizing spatial modes. This extension allows for predicting and designing the shape of the concentration gradient without simulating the governing equations. The streamlined analysis process of self-organized pattern formation is demonstrated with a simple illustrative reaction model with diffusion and advection.
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Affiliation(s)
- Yutaka Hori
- Department of Applied Physics and Physico-Informatics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan
| | - Hiroki Miyazako
- Department of Information Physics and Computing, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
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4
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Influence of fast advective flows on pattern formation of Dictyostelium discoideum. PLoS One 2018; 13:e0194859. [PMID: 29590179 PMCID: PMC5874059 DOI: 10.1371/journal.pone.0194859] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/30/2017] [Accepted: 03/12/2018] [Indexed: 12/05/2022] Open
Abstract
We report experimental and numerical results on pattern formation of self-organizing Dictyostelium discoideum cells in a microfluidic setup under a constant buffer flow. The external flow advects the signaling molecule cyclic adenosine monophosphate (cAMP) downstream, while the chemotactic cells attached to the solid substrate are not transported with the flow. At high flow velocities, elongated cAMP waves are formed that cover the whole length of the channel and propagate both parallel and perpendicular to the flow direction. While the wave period and transverse propagation velocity are constant, parallel wave velocity and the wave width increase linearly with the imposed flow. We also observe that the acquired wave shape is highly dependent on the wave generation site and the strength of the imposed flow. We compared the wave shape and velocity with numerical simulations performed using a reaction-diffusion model and found excellent agreement. These results are expected to play an important role in understanding the process of pattern formation and aggregation of D. discoideum that may experience fluid flows in its natural habitat.
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5
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Zhao J, Tian JP, Wei J. Minimal Model of Plankton Systems Revisited with Spatial Diffusion and Maturation Delay. Bull Math Biol 2016; 78:381-412. [DOI: 10.1007/s11538-016-0147-3] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2015] [Accepted: 02/12/2016] [Indexed: 10/22/2022]
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6
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Parshad RD, Kumari N, Kasimov AR, Abderrahmane HA. Turing patterns and long-time behavior in a three-species food-chain model. Math Biosci 2014; 254:83-102. [PMID: 24952324 DOI: 10.1016/j.mbs.2014.06.007] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/12/2013] [Revised: 06/04/2014] [Accepted: 06/05/2014] [Indexed: 11/30/2022]
Abstract
We consider a spatially explicit three-species food chain model, describing generalist top predator-specialist middle predator-prey dynamics. We investigate the long-time dynamics of the model and show the existence of a finite dimensional global attractor in the product space, L(2)(Ω). We perform linear stability analysis and show that the model exhibits the phenomenon of Turing instability, as well as diffusion induced chaos. Various Turing patterns such as stripe patterns, mesh patterns, spot patterns, labyrinth patterns and weaving patterns are obtained, via numerical simulations in 1d as well as in 2d. The Turing and non-Turing space, in terms of model parameters, is also explored. Finally, we use methods from nonlinear time series analysis to reconstruct a low dimensional chaotic attractor of the model, and estimate its fractal dimension. This provides a lower bound, for the fractal dimension of the attractor, of the spatially explicit model.
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Affiliation(s)
- Rana D Parshad
- Department of Mathematics, Clarkson University, Potsdam, NY 13699, USA
| | - Nitu Kumari
- Department of Mathematics, Clarkson University, Potsdam, NY 13699, USA; School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175001, India.
| | - Aslan R Kasimov
- Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia
| | - Hamid Ait Abderrahmane
- Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia
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7
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A mathematical biologist's guide to absolute and convective instability. Bull Math Biol 2013; 76:1-26. [PMID: 24272387 DOI: 10.1007/s11538-013-9911-9] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2013] [Accepted: 10/08/2013] [Indexed: 10/26/2022]
Abstract
Mathematical models have been highly successful at reproducing the complex spatiotemporal phenomena seen in many biological systems. However, the ability to numerically simulate such phenomena currently far outstrips detailed mathematical understanding. This paper reviews the theory of absolute and convective instability, which has the potential to redress this inbalance in some cases. In spatiotemporal systems, unstable steady states subdivide into two categories. Those that are absolutely unstable are not relevant in applications except as generators of spatial or spatiotemporal patterns, but convectively unstable steady states can occur as persistent features of solutions. The authors explain the concepts of absolute and convective instability, and also the related concepts of remnant and transient instability. They give examples of their use in explaining qualitative transitions in solution behaviour. They then describe how to distinguish different types of instability, focussing on the relatively new approach of the absolute spectrum. They also discuss the use of the theory for making quantitative predictions on how spatiotemporal solutions change with model parameters. The discussion is illustrated throughout by numerical simulations of a model for river-based predator-prey systems.
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8
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Ebeling W, Feistel R, Feistel S. On bifurcations in complex ecological systems with diffusion and noise. ECOLOGICAL COMPLEXITY 2013. [DOI: 10.1016/j.ecocom.2012.09.007] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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9
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Haque M. Existence of complex patterns in the Beddington–DeAngelis predator–prey model. Math Biosci 2012; 239:179-90. [DOI: 10.1016/j.mbs.2012.05.006] [Citation(s) in RCA: 35] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/13/2011] [Revised: 05/10/2012] [Accepted: 05/16/2012] [Indexed: 11/15/2022]
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10
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Kurushina SE, Maximov VV, Romanovskii YM. Spatial pattern formation in external noise: theory and simulation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:011124. [PMID: 23005385 DOI: 10.1103/physreve.86.011124] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/01/2012] [Revised: 04/12/2012] [Indexed: 06/01/2023]
Abstract
Spatial pattern formation in fluctuating media is researched analytically from the point of view of the order parameters concept. A reaction-diffusion system with external noise is considered as a model of such media. Stochastic equations for unstable mode amplitudes (order parameters), the dispersion equation for averaged amplitudes of unstable modes, and the Fokker-Planck equation for the order parameters are obtained. The theory developed makes it possible to analyze different noise-induced effects including the variation of boundaries of ordering and disordering phase transitions depending on the parameters of external noise.
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Affiliation(s)
- S E Kurushina
- Physics Department, Samara State Aerospace University, Moskovskoye Shosse 34, Samara 443086, Russian Federation.
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11
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Upadhyay RK, Volpert V, Thakur NK. Propagation of Turing patterns in a plankton model. JOURNAL OF BIOLOGICAL DYNAMICS 2012; 6:524-538. [PMID: 22873604 DOI: 10.1080/17513758.2012.655327] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
The paper is devoted to a reaction-diffusion system of equations describing phytoplankton and zooplankton distributions. Linear stability analysis of the model is carried out. Turing and Hopf stability boundaries are found. Emergence of two-dimensional spatial structures is illustrated by numerical simulations. Travelling waves between various stationary solutions are investigated. Transitions between homogeneous in space stationary solutions and Turing structures are studied.
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Affiliation(s)
- R K Upadhyay
- Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand, 826004, India
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12
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Impact of Directed Movement on Invasive Spread in Periodic Patchy Environments. Bull Math Biol 2012; 74:1448-67. [DOI: 10.1007/s11538-011-9714-9] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/22/2011] [Accepted: 12/15/2011] [Indexed: 10/14/2022]
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13
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Sun GQ, Jin Z, Liu QX, Li BL. Rich dynamics in a predator–prey model with both noise and periodic force. Biosystems 2010; 100:14-22. [DOI: 10.1016/j.biosystems.2009.12.003] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2009] [Revised: 10/18/2009] [Accepted: 12/02/2009] [Indexed: 11/25/2022]
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14
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Lee JM, Hillen T, Lewis MA. Pattern formation in prey-taxis systems. JOURNAL OF BIOLOGICAL DYNAMICS 2009; 3:551-573. [PMID: 22880961 DOI: 10.1080/17513750802716112] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
In this paper, we consider spatial predator-prey models with diffusion and prey-taxis. We investigate necessary conditions for pattern formation using a variety of non-linear functional responses, linear and non-linear predator death terms, linear and non-linear prey-taxis sensitivities, and logistic growth or growth with an Allee effect for the prey. We identify combinations of the above non-linearities that lead to spatial pattern formation and we give numerical examples. It turns out that prey-taxis stabilizes the system and for large prey-taxis sensitivity we do not observe pattern formation. We also study and find necessary conditions for global stability for a type I functional response, logistic growth for the prey, non-linear predator death terms, and non-linear prey-taxis sensitivity.
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Affiliation(s)
- J M Lee
- Centre for Mathematical Biology, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1.
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15
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Bailey SF, McCauley E. Extrinsically and intrinsically generated spatial patterns of algal abundance in an experimental stream. ECOLOGICAL COMPLEXITY 2009. [DOI: 10.1016/j.ecocom.2009.01.001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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16
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Transient Responses to Spatial Perturbations in Advective Systems. Bull Math Biol 2008; 70:1480-502. [DOI: 10.1007/s11538-008-9309-2] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/05/2007] [Accepted: 01/25/2008] [Indexed: 10/22/2022]
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17
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Baurmann M, Gross T, Feudel U. Instabilities in spatially extended predator–prey systems: Spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations. J Theor Biol 2007; 245:220-9. [DOI: 10.1016/j.jtbi.2006.09.036] [Citation(s) in RCA: 227] [Impact Index Per Article: 13.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/20/2006] [Revised: 08/04/2006] [Accepted: 09/19/2006] [Indexed: 10/24/2022]
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18
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Spatiotemporal patterns in an excitable plankton system with lysogenic viral infection. ACTA ACUST UNITED AC 2005. [DOI: 10.1016/j.mcm.2004.10.025] [Citation(s) in RCA: 40] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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19
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20
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Sapoukhina N, Tyutyunov Y, Arditi R. The role of prey taxis in biological control: a spatial theoretical model. Am Nat 2003; 162:61-76. [PMID: 12856237 DOI: 10.1086/375297] [Citation(s) in RCA: 67] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2002] [Accepted: 10/22/2002] [Indexed: 11/03/2022]
Abstract
We study a reaction-diffusion-advection model for the dynamics of populations under biological control. A control agent is assumed to be a predator species that has the ability to perceive the heterogeneity of pest distribution. The advection term represents the predator density movement according to a basic prey taxis assumption: acceleration of predators is proportional to the prey density gradient. The prey population reproduces logistically, and the local population interactions follow the Holling Type II trophic function. On the scale of the population, our spatially explicit approach subdivides the predation process into random movement represented by diffusion, directed movement described by prey taxis, local prey encounters, and consumption modeled by the trophic function. Thus, our model allows studying the effects of large-scale predator spatial activity on population dynamics. We show under which conditions spatial patterns are generated by prey taxis and how this affects the predator ability to maintain the pest population below some economic threshold. In particular, intermediate taxis activity can stabilize predator-pest populations at a very low level of pest density, ensuring successful biological control. However, very intensive prey taxis destroys the stability, leading to chaotic dynamics with pronounced outbreaks of pest density.
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Affiliation(s)
- Natalia Sapoukhina
- Laboratory of Mathematical Modelling of Biological Processes, Vorovich Research Institute of Mechanics and Applied Mathematics, Rostov State University, 200/1 Stachki Street, 344090 Rostov-on-Don, Russia.
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Malchow H, Petrovskii SV, Medvinsky AB. Numerical study of plankton–fish dynamics in a spatially structured and noisy environment. Ecol Modell 2002. [DOI: 10.1016/s0304-3800(01)00467-7] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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22
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Arditi R, Tyutyunov Y, Morgulis A, Govorukhin V, Senina I. Directed movement of predators and the emergence of density-dependence in predator-prey models. Theor Popul Biol 2001; 59:207-21. [PMID: 11444960 DOI: 10.1006/tpbi.2001.1513] [Citation(s) in RCA: 63] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
Abstract
We consider a bitrophic spatially distributed community consisting of prey and actively moving predators. The model is based on the assumption that the spatial and temporal variations of the predators' velocity are determined by the prey gradient. Locally, the populations follow the simple Lotka-Volterra interaction. We also assume predator reproduction and mortality to be negligible in comparison with the time scale of migration. The model demonstrates heterogeneous oscillating distributions of both species, which occur because of the active movements of predators. One consequence of this heterogeneity is increased viability of the prey population, compared to the equivalent homogeneous model, and increased consumption. Further numerical analysis shows that, on the spatially aggregated scale, the average predator density adversely affects the individual consumption, leading to a nonlinear predator-dependent trophic function, completely different from the Lotka-Volterra rule assumed at the local scale.
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Affiliation(s)
- R Arditi
- Ecologie des populations et communautés, Institut national agronomique Paris-Grignon, 75231 Paris cedex 05, France
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23
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Petrovskii SV, Malchow H. Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics. Theor Popul Biol 2001; 59:157-74. [PMID: 11302760 DOI: 10.1006/tpbi.2000.1509] [Citation(s) in RCA: 172] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
Abstract
The dynamics of a simple prey-predator system is described by a system of two reaction- diffusion equations with biologically reasonable non-linearities (logistic growth of the prey, Holling type II functional response of the predator). We show that, when the local kinetics of the system is oscillatory, for a wide class of initial conditions the evolution of the system leads to the formation of a non-stationary irregular pattern corresponding to spatio-temporal chaos. The chaotic pattern first appears inside a sub-domain of the system. This sub-domain then steadily grows with time and, finally, the chaotic pattern invades the whole space, displacing the regular pattern.
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Affiliation(s)
- S V Petrovskii
- Shirshov Institute of Oceanology, Russian Academy of Sciences, Nakhimovsky Prospect 36, Moscow, 117218, Russia.
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