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A two-patch prey-predator model with predator dispersal driven by the predation strength

1. Sciences and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA
2. Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India
3. Simon A. Levin Mathematical and Computational Modeling Sciences Center, Arizona State University, Mesa, AZ 85212, USA

Foraging movements of predator play an important role in population dynamics of prey-predator systems, which have been considered as mechanisms that contribute to spatial self-organization of prey and predator. In nature, there are many examples of prey-predator interactions where prey is immobile while predator disperses between patches non-randomly through different factors such as stimuli following the encounter of a prey. In this work, we formulate a Rosenzweig-MacArthur prey-predator two patch model with mobility only in predator and the assumption that predators move towards patches with more concentrated prey-predator interactions. We provide completed local and global analysis of our model. Our analytical results combined with bifurcation diagrams suggest that: (1) dispersal may stabilize or destabilize the coupled system; (2) dispersal may generate multiple interior equilibria that lead to rich bistable dynamics or may destroy interior equilibria that lead to the extinction of predator in one patch or both patches; (3) Under certain conditions, the large dispersal can promote the permanence of the system. In addition, we compare the dynamics of our model to the classic two patch model to obtain a better understanding how different dispersal strategies may have different impacts on the dynamics and spatial patterns.

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Keywords Rosenzweig-MacArthur prey-predator model; self-organization effects; dispersal; persistence; non-random foraging movements

Citation: Yun Kang, Sourav Kumar Sasmal, Komi Messan. A two-patch prey-predator model with predator dispersal driven by the predation strength. Mathematical Biosciences and Engineering, 2017, 14(4): 843-880. doi: 10.3934/mbe.2017046


  • [1] L. Aarssen,R. Turkington, Biotic specialization between neighbouring genotypes in lolium perenne and trifolium repens from a permanent pasture, The Journal of Ecology, 73 (1985): 605-614.
  • [2] R. F. Alder, Migration alone can produce persistence of host-parasitoid models, The American Naturalist, 141 (1993): 642-650.
  • [3] J. Bascompte,R. V. Solé, Spatially induced bifurcations in single-species population dynamics, Journal of Animal Ecology, 63 (1994): 256-264.
  • [4] B. M. Bolker,S. W. Pacala, Spatial moment equations for plant competition: Understanding spatial strategies and the advantages of short dispersal, The American Naturalist, 153 (1999): 575-602.
  • [5] C. J. Bolter,M. Dicke,J. J. Van Loon,J. Visser,M. A. Posthumus, Attraction of colorado potato beetle to herbivore-damaged plants during herbivory and after its termination, Journal of Chemical Ecology, 23 (1997): 1003-1023.
  • [6] C. Carroll,D. H. Janzen, Ecology of foraging by ants, Annual Review of Ecology and Systematics, 4 (1973): 231-257.
  • [7] A. Casal,J. Eilbeck,J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Differential and Integral Equations, 7 (1994): 411-439.
  • [8] P. L. Chesson,W. W. Murdoch, Aggregation of risk: Relationships among host-parasitoid models, American Naturalist, 127 (1986): 696-715.
  • [9] W. C. Chewning, Migratory effects in predator-prey models, Mathematical Biosciences, 23 (1975): 253-262.
  • [10] R. Cressman,K. Vlastimil, Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds, Journal of Mathematical Biology, 67 (2013): 329-358.
  • [11] E. Curio, The Ethology of Predation ,Springer-Verlag Berlin Heidelberg, 7 1976.
  • [12] M. Doebli, Dispersal and dynamics, Theoretical Population Biology, 47 (1995): 82-106.
  • [13] W. Feng,B. Rock,J. Hinson, On a new model of two-patch predator-prey system with migration of both species, Journal of Applied Analysis and Computation, 1 (2011): 193-203.
  • [14] J. Ford, The Role of the Trypanosomiases in African Ecology. A Study of the Tsetse Fly Problem, in Oxford University Press, Oxford, 1971.
  • [15] A. G. Gatehouse, Permanence and the dynamics of biological systems, Host Finding Behaviour Of Tsetse Flies, null (1972): 83-95.
  • [16] S. Ghosh,S. Bhattacharyya, A two-patch prey-predator model with food-gathering activity, Journal of Applied Mathematics and Computing, 37 (2011): 497-521.
  • [17] M. Gillies,T. Wilkes, The range of attraction of single baits for some West African mosquitoes, Bulletin of Entomological Research, 60 (1970): 225-235.
  • [18] M. Gillies,T. Wilkes, The range of attraction of animal baits and carbon dioxide for mosquitoes, Bulletin of Entomological Research, 61 (1972): 389-404.
  • [19] M. Gillies,T. Wilkes, The range of attraction of birds as baits for some west african mosquitoes (diptera, culicidae), Bulletin of Entomological Research, 63 (1974): 573-582.
  • [20] I. Hanski, null, Metapopulation Ecology, Oxford University Press, Oxford, 1999.
  • [21] I. A. Hanski,M. E. Gilpin, null, Metapopulation Biology: Ecology, Genetics, and Evolution, Academic Press, San Diego, 1997.
  • [22] M. Hassell,R. May, Aggregation of predators and insect parasites and its effect on stability, The Journal of Animal Ecology, 43 (1974): 567-594.
  • [23] M. Hassell,T. Southwood, Foraging strategies of insects, Annual Review of Ecology and Systematics, 9 (1978): 75-98.
  • [24] M. Hassell,O. Miramontes,P. Rohani,R. May, Appropriate formulations for dispersal in spatially structured models: comments on bascompte & Solé, Journal of Animal Ecology, 64 (1995): 662-664.
  • [25] M. P. Hassell,H. N. Comins,R. M. May, Spatial structure and chaos in insect population dynamics, Nature, 353 (1991): 255-258.
  • [26] A. Hastings, Can spatial variation along lead to selection for dispersal?, Theoretical Population Biology, 24 (1983): 244-251.
  • [27] A. Hastings, Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations, Ecology, 74 (1993): 1362-1372.
  • [28] C. Hauzy,M. Gauduchon,F. D. Hulot,M. Loreau, Density-dependent dispersal and relative dispersal affect the stability of predator-prey metacommunities, Journal of Theoretical Biology, 266 (2010): 458-469.
  • [29] R. D. Holt, Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution, Theoretical Population Biology, 28 (1985): 181-208.
  • [30] S. Hsu,S. Hubbell,P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977): 366-383.
  • [31] S. Hsu, On global stability of a predator-prey system, Mathematical Biosciences, 39 (1978): 1-10.
  • [32] Y. Huang,O. Diekmann, Predator migration in response to prey density: What are the consequences?, Journal of Mathematical Biology, 43 (2001): 561-581.
  • [33] V. Hutson, A theorem on average liapunov functions, Monatshefte für Mathematik, 98 (1984): 267-275.
  • [34] V. Hutson,K. Schmit, Permanence and the dynamics of biological systems, Mathematical Biosciences, 111 (1992): 1-71.
  • [35] V. A. Jansen, Regulation of predator-prey systems through spatial interactions: A possible solution to the paradox of enrichment, Oikos, 74 (1995): 384-390.
  • [36] V. A. Jansen, The dynamics of two diffusively coupled predator-prey populations, Theoretical Population Biology, 59 (2001): 119-131.
  • [37] V. A. A. Jansen, Theoretical Aspects of Metapopulation Dynamics, PhD thesis, Ph. D. thesis, Leiden University, The Netherlands, 1994.
  • [38] Y. Kang,D. Armbruster, Dispersal effects on a discrete two-patch model for plant-insect interactions, Journal of Theoretical Biology, 268 (2011): 84-97.
  • [39] Y. Kang,C. Castillo-Chavez, Multiscale analysis of compartment models with dispersal, Journal of Biological Dynamics, 6 (2012): 50-79.
  • [40] P. Kareiva,G. Odell, Swarms of predators exhibit "prey-taxis" if individual predators use area-restricted search, American Naturalist,, 130 (1987): 233-270.
  • [41] P. Kareiva,A. Mullen,R. Southwood, Population dynamics in spatially complex environments: Theory and data [and discussion], Philosophical Transactions of the Royal Society of London B: Biological Sciences, 330 (1990): 175-190.
  • [42] S. Kéfi,M. Rietkerk,M. van Baalen,M. Loreau, Local facilitation, bistability and transitions in arid ecosystems, Theoretical Population Biology, 71 (2007): 367-379.
  • [43] P. Klepac,M. G. Neubert,P. van den Driessche, Dispersal delays, predator-prey stability, and the paradox of enrichment, Theoretical Population Biology, 71 (2007): 436-444.
  • [44] M. Kummel,D. Brown,A. Bruder, How the aphids got their spots: Predation drives self-organization of aphid colonies in a patchy habitat, Oikos, 122 (2013): 896-906.
  • [45] K. Kuto,Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, Journal of Differential Equations, 197 (2004): 315-348.
  • [46] I. Lengyel,I. R. Epstein, Diffusion-induced instability in chemically reacting systems: Steady-state multiplicity, oscillation, and chaos, Chaos: An Interdisciplinary Journal of Nonlinear Science, 1 (1991): 69-76.
  • [47] S. A. Levin, Dispersion and population interactions, American Naturalist, 108 (1974): 207-228.
  • [48] R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bulletin of the Entomological Society of America, 15 (1969): 237-240.
  • [49] Z.-z. Li,M. Gao,C. Hui,X.-z. Han,H. Shi, Impact of predator pursuit and prey evasion on synchrony and spatial patterns in metapopulation, Ecological Modelling, 185 (2005): 245-254.
  • [50] X. Liu,L. Chen, Complex dynamics of Holling type Ⅱ Lotka--Volterra predator--prey system with impulsive perturbations on the predator, Chaos, Solitons & Fractals, 16 (2003): 311-320.
  • [51] Y. Liu, The Dynamical Behavior of a Two Patch Predator-Prey Model ,Honor Thesis, from The College of William and Mary, 2010.
  • [52] J. H. Loughrin,D. A. Potter,T. R. Hamilton-Kemp,M. E. Byers, Role of feeding-induced plant volatiles in aggregative behavior of the japanese beetle (coleoptera: Scarabaeidae), Environmental Entomology, 25 (1996): 1188-1191.
  • [53] J. Madden, Physiological reactions of Pinus radiata to attack by woodwasp, Sirex noctilio F.(Hymenoptera: Siricidae), Bulletin of Entomological Research, 67 (1977): 405-426.
  • [54] L. Markus, Ⅱ. Asymptotically autonomous differential systems, in Contributions to the Theory of Nonlinear Oscillations (AM-36), Vol. Ⅲ, Princeton University Press, 1956, 17–30.
  • [55] R. M. May, Host-parasitoid systems in patchy environments: A phenomenological model, The Journal of Animal Ecology, 47 (1978): 833-844.
  • [56] R. McMurtrie, Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments, Mathematical Biosciences, 39 (1978): 11-51.
  • [57] T. F. Miller,D. J. Mladenoff,M. K. Clayton, Old-growth northern hardwood forests: Spatial autocorrelation and patterns of understory vegetation, Ecological Monographs, 72 (2002): 487-503.
  • [58] W. W. Murdoch,C. J. Briggs,R. M. Nisbet,W. S. Gurney,A. Stewart-Oaten, Aggregation and stability in metapopulation models, American Naturalist, 140 (1992): 41-58.
  • [59] M. Pascual, Diffusion-induced chaos in a spatial predator-prey system, Proceedings of the Royal Society of London B: Biological Sciences, 251 (1993): 1-7.
  • [60] M. Rees,P. J. Grubb,D. Kelly, Quantifying the impact of competition and spatial heterogeneity on the structure and dynamics of a four-species guild of winter annuals, American Naturalist, 147 (1996): 1-32.
  • [61] M. Rietkerk,J. Van de Koppel, Regular pattern formation in real ecosystems, Trends in Ecology & Evolution, 23 (2008): 169-175.
  • [62] P. Rohani,G. D. Ruxton, Dispersal and stability in metapopulations, Mathematical Medicine and Biology, 16 (1999): 297-306.
  • [63] M. L. Rosenzweig,R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963): 209-223.
  • [64] G. D. Ruxton, Density-dependent migration and stability in a system of linked populations, Bulletin of Mathematical Biology, 58 (1996): 643-660.
  • [65] L. M. Schoonhoven, Plant recognition by lepidopterous larvae, (1972), 87–99.
  • [66] L. M. Schoonhoven, On the variability of chemosensory information, The Host-Plant in Relation to Insect Behaviour and Reproduction, Symp. Biol. Hung., 16 (1976), 261–266.
  • [67] L. M. Schoonhoven, Chemosensory systems and feeding behavior in phytophagous insects, (1977), 391–398.
  • [68] E. W. Seabloom,O. N. Bjørnstad,B. M. Bolker,O. Reichman, Spatial signature of environmental heterogeneity, dispersal, and competition in successional grasslands, Ecological Monographs, 75 (2005): 199-214.
  • [69] G. Seifert and L. Markus, Contributions to the Theory of Nonlinear Oscillations ,Princeton University Press, 1956.
  • [70] Y. Shahak,E. Gal,Y. Offir,D. Ben-Yakir, Photoselective shade netting integrated with greenhouse technologies for improved performance of vegetable and ornamental crops, International Workshop on Greenhouse Environmental Control and Crop Production in Semi-Arid Regions, 797 (2008): 75-80.
  • [71] R. V. Solé and J. Bascompte, Self-Organization in Complex Ecosystems ,Princeton University Press, Princeton, 2006.
  • [72] A. Soro,S. Sundberg,H. Rydin, Species diversity, niche metrics and species associations in harvested and undisturbed bogs, Journal of Vegetation Science, 10 (1999): 549-560.
  • [73] H. R. Thieme, Mathematics in Population Biology ,Princeton University Press, 2003.
  • [74] D. Tilman and P. M. Kareiva, Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions ,volume 30, Princeton University Press, 1997.
  • [75] J. van de Koppel,J. C. Gascoigne,G. Theraulaz,M. Rietkerk,W. M. Mooij,P. M. Herman, Experimental evidence for spatial self-organization and its emergent effects in mussel bed ecosystems, Science, 322 (2008): 739-742.
  • [76] J. K. Waage, Behavioral Aspects of Foraging in the Parasitoid, Nemeritis Canescens (Grav. ) ,PhD Thesis, from University of London, 1977.
  • [77] J. Wang,J. Shi,J. Wei, Predator-prey system with strong Allee effect in prey, Journal of Mathematical Biology, 62 (2011): 291-331.


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