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Jorge DCP, Martinez-Garcia R. Demographic effects of aggregation in the presence of a component Allee effect. J R Soc Interface 2024; 21:20240042. [PMID: 38916901 DOI: 10.1098/rsif.2024.0042] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2023] [Accepted: 03/12/2024] [Indexed: 06/26/2024] Open
Abstract
The component Allee effect (AE) is the positive correlation between an organism's fitness component and population density. Depending on the population spatial structure, which determines the interactions between organisms, a component AE might lead to positive density dependence in the population per-capita growth rate and establish a demographic AE. However, existing spatial models impose a fixed population spatial structure, which limits the understanding of how a component AE and spatial dynamics jointly determine the existence of demographic AEs. We introduce a spatially explicit theoretical framework where spatial structure and population dynamics are emergent properties of the individual-level demographic and movement rates. This framework predicts various spatial patterns depending on its specific parametrization, including evenly spaced aggregates of organisms, which determine the demographic-level by-products of the component AE. We find that aggregation increases population abundance and allows population survival in harsher environments and at lower global population densities when compared with uniformly distributed organisms. Moreover, aggregation can prevent the component AE from manifesting at the population level or restrict it to the level of each independent aggregate. These results provide a mechanistic understanding of how component AEs might operate for different spatial structures and manifest at larger scales.
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Affiliation(s)
- Daniel C P Jorge
- ICTP South American Institute for Fundamental Research & Instituto de Física Teórica, Universidade Estadual Paulista-UNESP, Rua Dr. Bento Teobaldo Ferraz 271, Bloco 2-Barra Funda , São Paulo, SP 01140-070, Brazil
- Department of Ecology and Evolutionary Biology, Princeton University , Princeton, NJ 08544, USA
| | - Ricardo Martinez-Garcia
- ICTP South American Institute for Fundamental Research & Instituto de Física Teórica, Universidade Estadual Paulista-UNESP, Rua Dr. Bento Teobaldo Ferraz 271, Bloco 2-Barra Funda , São Paulo, SP 01140-070, Brazil
- Center for Advanced Systems Understanding (CASUS), Helmholtz-Zentrum Dresden Rossendorf (HZDR) , Görlitz 02826, Germany
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Chen M, Xu Y, Zhao J, Wei X. Turing-Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator-Prey Model with Allee Effect and Predator Harvesting. ENTROPY (BASEL, SWITZERLAND) 2023; 26:18. [PMID: 38248144 PMCID: PMC10814816 DOI: 10.3390/e26010018] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2023] [Revised: 12/17/2023] [Accepted: 12/20/2023] [Indexed: 01/23/2024]
Abstract
This paper investigates the complex dynamics of a ratio-dependent predator-prey model incorporating the Allee effect in prey and predator harvesting. To explore the joint effect of the harvesting effort and diffusion on the dynamics of the system, we perform the following analyses: (a) The stability of non-negative constant steady states; (b) The sufficient conditions for the occurrence of a Hopf bifurcation, Turing bifurcation, and Turing-Hopf bifurcation; (c) The derivation of the normal form near the Turing-Hopf singularity. Moreover, we provide numerical simulations to illustrate the theoretical results. The results demonstrate that the small change in harvesting effort and the ratio of the diffusion coefficients will destabilize the constant steady states and lead to the complex spatiotemporal behaviors, including homogeneous and inhomogeneous periodic solutions and nonconstant steady states. Moreover, the numerical simulations coincide with our theoretical results.
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Affiliation(s)
| | | | | | - Xin Wei
- Ecological Restoration and Resource Utilization for Cold Region, School of Mathematical Science, Heilongjiang University, Harbin 150080, China; (M.C.); (Y.X.); (J.Z.)
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Turing Instability and Spatiotemporal Pattern Formation Induced by Nonlinear Reaction Cross-Diffusion in a Predator–Prey System with Allee Effect. MATHEMATICS 2022. [DOI: 10.3390/math10091500] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
The Allee effect is widespread among endangered plants and animals in ecosystems, suggesting that a minimum population density or size is necessary for population survival. This paper investigates the stability and pattern formation of a predator–prey model with nonlinear reactive cross-diffusion under Neumann boundary conditions, which introduces the Allee effect. Firstly, the ODE system is asymptotically stable for its positive equilibrium solution. In a reaction system with self-diffusion, the Allee effect can destabilize the system. Then, in a reaction system with cross-diffusion, through a linear stability analysis, the cross-diffusion coefficient is used as a bifurcation parameter, and instability conditions driven by the cross-diffusion are obtained. Furthermore, we show that the system (5) has at least one inhomogeneous stationary solution. Finally, our theoretical results are illustrated with numerical simulations.
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Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion. MATHEMATICS 2019. [DOI: 10.3390/math7121172] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
In this work, we investigate numerically a system of partial differential equations that describes the interactions between populations of predators and preys. The system considers the effects of anomalous diffusion and generalized Michaelis–Menten-type reactions. For the sake of generality, we consider an extended form of that system in various spatial dimensions and propose two finite-difference methods to approximate its solutions. Both methodologies are presented in alternative forms to facilitate their analyses and computer implementations. We show that both schemes are structure-preserving techniques, in the sense that they can keep the positive and bounded character of the computational approximations. This is in agreement with the relevant solutions of the original population model. Moreover, we prove rigorously that the schemes are consistent discretizations of the generalized continuous model and that they are stable and convergent. The methodologies were implemented efficiently using MATLAB. Some computer simulations are provided for illustration purposes. In particular, we use our schemes in the investigation of complex patterns in some two- and three-dimensional predator–prey systems with anomalous diffusion.
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Yu T, Tian Y, Guo H, Song X. Dynamical analysis of an integrated pest management predator-prey model with weak Allee effect. JOURNAL OF BIOLOGICAL DYNAMICS 2019; 13:218-244. [PMID: 30887880 DOI: 10.1080/17513758.2019.1589000] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2018] [Accepted: 02/21/2019] [Indexed: 06/09/2023]
Abstract
In this paper, a pest management predator-prey model with weak Allee effect on predator and state feedback impulsive control on prey is introduced and analysed, where the yield of predator released and intensity of pesticide sprayed are assumed to be linearly dependent on the selected pest control level. For the proposed model, the existence and stability of the order-1 periodic orbit of the control system are discussed. Meanwhile, with the aim of minimizing the input cost in practice, an optimization model is constructed to determine the optimal quantity of the predator released and the intensity of pesticide sprayed. The theoretical results and numerical simulations indicated that the number of pests can be limited to below an economic threshold and displays periodic variation under the proposed control strategy. In addition, it indicated in numerical simulations that an order-2 periodic orbit exists for some certain parameters.
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Affiliation(s)
- Tingting Yu
- a School of Mathematics and Statistics, Xinyang Normal University , Xinyang , People's Republic of China
- b Business School, University of Shanghai for Science and Technology , Shanghai , People's Republic of China
| | - Yuan Tian
- c School of Science, Dalian Maritime University , Dalian , People's Republic of China
| | - Hongjian Guo
- a School of Mathematics and Statistics, Xinyang Normal University , Xinyang , People's Republic of China
| | - Xinyu Song
- a School of Mathematics and Statistics, Xinyang Normal University , Xinyang , People's Republic of China
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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.
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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
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SINGH TEEKAM, BANERJEE SANDIP. SPATIAL ASPECT OF HUNTING COOPERATION IN PREDATORS WITH HOLLING TYPE II FUNCTIONAL RESPONSE. J BIOL SYST 2019. [DOI: 10.1142/s0218339018500237] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we have investigated a spatial predator–prey model with hunting cooperation in predators. Using linear stability analysis, we obtain the condition for diffusive instability and identify the corresponding domain in the space of controlling parameters. Using extensive numerical simulations, we obtain complex patterns, namely, spotted pattern, stripe pattern and mixed pattern in the Turing domain, by varying the hunting cooperation parameter in predators and carrying capacity of preys. The results focus on the effect of hunting cooperation in pattern dynamics of a diffusive predator–prey model and help us in better understanding of the dynamics of the predator–prey interaction in real environments.
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Affiliation(s)
- TEEKAM SINGH
- Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667, India
| | - SANDIP BANERJEE
- Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667, India
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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]
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