
-
Previous Article
A proton therapy model using discrete difference equations with an example of treating hepatocellular carcinoma
- MBE Home
- This Issue
-
Next Article
A chaotic bursting-spiking transition in a pancreatic beta-cells system: observation of an interior glucose-induced crisis
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.
References:
[1] |
L. Aarssen and 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.
doi: 10.2307/2260497. |
[2] |
R. F. Alder,
Migration alone can produce persistence of host-parasitoid models, The American Naturalist, 141 (1993), 642-650.
|
[3] |
J. Bascompte and R. V. Solé,
Spatially induced bifurcations in single-species population dynamics, Journal of Animal Ecology, 63 (1994), 256-264.
doi: 10.2307/5544. |
[4] |
B. M. Bolker and 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.
doi: 10.1086/303199. |
[5] |
C. J. Bolter, M. Dicke, J. J. Van Loon, J. Visser and 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.
doi: 10.1023/B:JOEC.0000006385.70652.5e. |
[6] |
C. Carroll and D. H. Janzen,
Ecology of foraging by ants, Annual Review of Ecology and Systematics, 4 (1973), 231-257.
doi: 10.1146/annurev.es.04.110173.001311. |
[7] |
A. Casal, J. Eilbeck and 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 and W. W. Murdoch,
Aggregation of risk: Relationships among host-parasitoid models, American Naturalist, 127 (1986), 696-715.
doi: 10.1086/284514. |
[9] |
W. C. Chewning,
Migratory effects in predator-prey models, Mathematical Biosciences, 23 (1975), 253-262.
doi: 10.1016/0025-5564(75)90039-5. |
[10] |
R. Cressman and K. Vlastimil,
Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds, Journal of Mathematical Biology, 67 (2013), 329-358.
doi: 10.1007/s00285-012-0548-3. |
[11] |
E. Curio,
The Ethology of Predation ,Springer-Verlag Berlin Heidelberg, 7 1976.
doi: 10.1007/978-3-642-81028-2. |
[12] |
M. Doebli,
Dispersal and dynamics, Theoretical Population Biology, 47 (1995), 82-106.
doi: 10.1006/tpbi.1995.1004. |
[13] |
W. Feng, B. Rock and 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, (1972), 83-95.
|
[16] |
S. Ghosh and S. Bhattacharyya,
A two-patch prey-predator model with food-gathering activity, Journal of Applied Mathematics and Computing, 37 (2011), 497-521.
doi: 10.1007/s12190-010-0446-z. |
[17] |
M. Gillies and T. Wilkes,
The range of attraction of single baits for some West African mosquitoes, Bulletin of Entomological Research, 60 (1970), 225-235.
doi: 10.1017/S000748530004075X. |
[18] |
M. Gillies and 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 and 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.
doi: 10.1017/S0007485300047817. |
[20] |
I. Hanski, Metapopulation Ecology, Oxford University Press, Oxford, 1999.
![]() |
[21] |
I. A. Hanski and M. E. Gilpin, Metapopulation Biology: Ecology, Genetics, and Evolution, Academic Press, San Diego, 1997.
![]() |
[22] |
M. Hassell and R. May,
Aggregation of predators and insect parasites and its effect on stability, The Journal of Animal Ecology, 43 (1974), 567-594.
doi: 10.2307/3384. |
[23] |
M. Hassell and T. Southwood,
Foraging strategies of insects, Annual Review of Ecology and Systematics, 9 (1978), 75-98.
doi: 10.1146/annurev.es.09.110178.000451. |
[24] |
M. Hassell, O. Miramontes, P. Rohani and R. May,
Appropriate formulations for dispersal in spatially structured models: comments on bascompte & Solé, Journal of Animal Ecology, 64 (1995), 662-664.
doi: 10.2307/5808. |
[25] |
M. P. Hassell, H. N. Comins and R. M. May,
Spatial structure and chaos in insect population dynamics, Nature, 353 (1991), 255-258.
doi: 10.1038/353255a0. |
[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.
doi: 10.2307/1940066. |
[28] |
C. Hauzy, M. Gauduchon, F. D. Hulot and M. Loreau,
Density-dependent dispersal and relative dispersal affect the stability of predator-prey metacommunities, Journal of Theoretical Biology, 266 (2010), 458-469.
doi: 10.1016/j.jtbi.2010.07.008. |
[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.
doi: 10.1016/0040-5809(85)90027-9. |
[30] |
S. Hsu, S. Hubbell and P. Waltman,
A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.
doi: 10.1137/0132030. |
[31] |
S. Hsu,
On global stability of a predator-prey system, Mathematical Biosciences, 39 (1978), 1-10.
doi: 10.1016/0025-5564(78)90025-1. |
[32] |
Y. Huang and O. Diekmann,
Predator migration in response to prey density: What are the consequences?, Journal of Mathematical Biology, 43 (2001), 561-581.
doi: 10.1007/s002850100107. |
[33] |
V. Hutson,
A theorem on average liapunov functions, Monatshefte für Mathematik, 98 (1984), 267-275.
doi: 10.1007/BF01540776. |
[34] |
V. Hutson and K. Schmit,
Permanence and the dynamics of biological systems, Mathematical Biosciences, 111 (1992), 1-71.
doi: 10.1016/0025-5564(92)90078-B. |
[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.
doi: 10.2307/3545983. |
[36] |
V. A. Jansen,
The dynamics of two diffusively coupled predator-prey populations, Theoretical Population Biology, 59 (2001), 119-131.
doi: 10.1006/tpbi.2000.1506. |
[37] |
V. A. A. Jansen,
Theoretical Aspects of Metapopulation Dynamics, PhD thesis, Ph. D. thesis, Leiden University, The Netherlands, 1994. |
[38] |
Y. Kang and D. Armbruster,
Dispersal effects on a discrete two-patch model for plant-insect interactions, Journal of Theoretical Biology, 268 (2011), 84-97.
doi: 10.1016/j.jtbi.2010.09.033. |
[39] |
Y. Kang and C. Castillo-Chavez,
Multiscale analysis of compartment models with dispersal, Journal of Biological Dynamics, 6 (2012), 50-79.
doi: 10.1080/17513758.2012.713125. |
[40] |
P. Kareiva and 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 and 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.
doi: 10.1098/rstb.1990.0191. |
[42] |
S. Kéfi, M. Rietkerk, M. van Baalen and M. Loreau,
Local facilitation, bistability and transitions in arid ecosystems, Theoretical Population Biology, 71 (2007), 367-379.
|
[43] |
P. Klepac, M. G. Neubert and P. van den Driessche,
Dispersal delays, predator-prey stability, and the paradox of enrichment, Theoretical Population Biology, 71 (2007), 436-444.
doi: 10.1016/j.tpb.2007.02.002. |
[44] |
M. Kummel, D. Brown and A. Bruder,
How the aphids got their spots: Predation drives self-organization of aphid colonies in a patchy habitat, Oikos, 122 (2013), 896-906.
doi: 10.1111/j.1600-0706.2012.20805.x. |
[45] |
K. Kuto and Y. Yamada,
Multiple coexistence states for a prey-predator system with cross-diffusion, Journal of Differential Equations, 197 (2004), 315-348.
doi: 10.1016/j.jde.2003.08.003. |
[46] |
I. Lengyel and 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.
doi: 10.1063/1.165819. |
[47] |
S. A. Levin,
Dispersion and population interactions, American Naturalist, 108 (1974), 207-228.
doi: 10.1086/282900. |
[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.
doi: 10.1093/besa/15.3.237. |
[49] |
Z.-z. Li, M. Gao, C. Hui, X.-z. Han and H. Shi,
Impact of predator pursuit and prey evasion on synchrony and spatial patterns in metapopulation, Ecological Modelling, 185 (2005), 245-254.
doi: 10.1016/j.ecolmodel.2004.12.008. |
[50] |
X. Liu and 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.
doi: 10.1016/S0960-0779(02)00408-3. |
[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 and M. E. Byers,
Role of feeding-induced plant volatiles in aggregative behavior of the japanese beetle (coleoptera: Scarabaeidae), Environmental Entomology, 25 (1996), 1188-1191.
doi: 10.1093/ee/25.5.1188. |
[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.
doi: 10.1017/S0007485300011214. |
[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.
doi: 10.1515/9781400882175-003. |
[55] |
R. M. May,
Host-parasitoid systems in patchy environments: A phenomenological model, The Journal of Animal Ecology, 47 (1978), 833-844.
doi: 10.2307/3674. |
[56] |
R. McMurtrie,
Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments, Mathematical Biosciences, 39 (1978), 11-51.
doi: 10.1016/0025-5564(78)90026-3. |
[57] |
T. F. Miller, D. J. Mladenoff and 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 and A. Stewart-Oaten,
Aggregation and stability in metapopulation models, American Naturalist, 140 (1992), 41-58.
doi: 10.1086/285402. |
[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.
doi: 10.1098/rspb.1993.0001. |
[60] |
M. Rees, P. J. Grubb and 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.
doi: 10.1086/285837. |
[61] |
M. Rietkerk and J. Van de Koppel,
Regular pattern formation in real ecosystems, Trends in Ecology & Evolution, 23 (2008), 169-175.
doi: 10.1016/j.tree.2007.10.013. |
[62] |
P. Rohani and G. D. Ruxton,
Dispersal and stability in metapopulations, Mathematical Medicine and Biology, 16 (1999), 297-306.
doi: 10.1093/imammb/16.3.297. |
[63] |
M. L. Rosenzweig and R. H. MacArthur,
Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223.
doi: 10.1086/282272. |
[64] |
G. D. Ruxton,
Density-dependent migration and stability in a system of linked populations, Bulletin of Mathematical Biology, 58 (1996), 643-660.
doi: 10.1007/BF02459477. |
[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.
doi: 10.1007/978-1-4613-4274-8_42. |
[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 and O. Reichman,
Spatial signature of environmental heterogeneity, dispersal, and competition in successional grasslands, Ecological Monographs, 75 (2005), 199-214.
doi: 10.1890/03-0841. |
[69] |
G. Seifert and L. Markus,
Contributions to the Theory of Nonlinear Oscillations ,Princeton University Press, 1956. |
[70] |
Y. Shahak, E. Gal, Y. Offir and 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.
doi: 10.17660/ActaHortic.2008.797.8. |
[71] |
R. V. Solé and J. Bascompte,
Self-Organization in Complex Ecosystems ,Princeton University Press, Princeton, 2006. |
[72] |
A. Soro, S. Sundberg and H. Rydin,
Species diversity, niche metrics and species associations in harvested and undisturbed bogs, Journal of Vegetation Science, 10 (1999), 549-560.
doi: 10.2307/3237189. |
[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 and 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 and J. Wei,
Predator-prey system with strong Allee effect in prey, Journal of Mathematical Biology, 62 (2011), 291-331.
doi: 10.1007/s00285-010-0332-1. |
show all references
References:
[1] |
L. Aarssen and 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.
doi: 10.2307/2260497. |
[2] |
R. F. Alder,
Migration alone can produce persistence of host-parasitoid models, The American Naturalist, 141 (1993), 642-650.
|
[3] |
J. Bascompte and R. V. Solé,
Spatially induced bifurcations in single-species population dynamics, Journal of Animal Ecology, 63 (1994), 256-264.
doi: 10.2307/5544. |
[4] |
B. M. Bolker and 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.
doi: 10.1086/303199. |
[5] |
C. J. Bolter, M. Dicke, J. J. Van Loon, J. Visser and 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.
doi: 10.1023/B:JOEC.0000006385.70652.5e. |
[6] |
C. Carroll and D. H. Janzen,
Ecology of foraging by ants, Annual Review of Ecology and Systematics, 4 (1973), 231-257.
doi: 10.1146/annurev.es.04.110173.001311. |
[7] |
A. Casal, J. Eilbeck and 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 and W. W. Murdoch,
Aggregation of risk: Relationships among host-parasitoid models, American Naturalist, 127 (1986), 696-715.
doi: 10.1086/284514. |
[9] |
W. C. Chewning,
Migratory effects in predator-prey models, Mathematical Biosciences, 23 (1975), 253-262.
doi: 10.1016/0025-5564(75)90039-5. |
[10] |
R. Cressman and K. Vlastimil,
Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds, Journal of Mathematical Biology, 67 (2013), 329-358.
doi: 10.1007/s00285-012-0548-3. |
[11] |
E. Curio,
The Ethology of Predation ,Springer-Verlag Berlin Heidelberg, 7 1976.
doi: 10.1007/978-3-642-81028-2. |
[12] |
M. Doebli,
Dispersal and dynamics, Theoretical Population Biology, 47 (1995), 82-106.
doi: 10.1006/tpbi.1995.1004. |
[13] |
W. Feng, B. Rock and 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, (1972), 83-95.
|
[16] |
S. Ghosh and S. Bhattacharyya,
A two-patch prey-predator model with food-gathering activity, Journal of Applied Mathematics and Computing, 37 (2011), 497-521.
doi: 10.1007/s12190-010-0446-z. |
[17] |
M. Gillies and T. Wilkes,
The range of attraction of single baits for some West African mosquitoes, Bulletin of Entomological Research, 60 (1970), 225-235.
doi: 10.1017/S000748530004075X. |
[18] |
M. Gillies and 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 and 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.
doi: 10.1017/S0007485300047817. |
[20] |
I. Hanski, Metapopulation Ecology, Oxford University Press, Oxford, 1999.
![]() |
[21] |
I. A. Hanski and M. E. Gilpin, Metapopulation Biology: Ecology, Genetics, and Evolution, Academic Press, San Diego, 1997.
![]() |
[22] |
M. Hassell and R. May,
Aggregation of predators and insect parasites and its effect on stability, The Journal of Animal Ecology, 43 (1974), 567-594.
doi: 10.2307/3384. |
[23] |
M. Hassell and T. Southwood,
Foraging strategies of insects, Annual Review of Ecology and Systematics, 9 (1978), 75-98.
doi: 10.1146/annurev.es.09.110178.000451. |
[24] |
M. Hassell, O. Miramontes, P. Rohani and R. May,
Appropriate formulations for dispersal in spatially structured models: comments on bascompte & Solé, Journal of Animal Ecology, 64 (1995), 662-664.
doi: 10.2307/5808. |
[25] |
M. P. Hassell, H. N. Comins and R. M. May,
Spatial structure and chaos in insect population dynamics, Nature, 353 (1991), 255-258.
doi: 10.1038/353255a0. |
[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.
doi: 10.2307/1940066. |
[28] |
C. Hauzy, M. Gauduchon, F. D. Hulot and M. Loreau,
Density-dependent dispersal and relative dispersal affect the stability of predator-prey metacommunities, Journal of Theoretical Biology, 266 (2010), 458-469.
doi: 10.1016/j.jtbi.2010.07.008. |
[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.
doi: 10.1016/0040-5809(85)90027-9. |
[30] |
S. Hsu, S. Hubbell and P. Waltman,
A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.
doi: 10.1137/0132030. |
[31] |
S. Hsu,
On global stability of a predator-prey system, Mathematical Biosciences, 39 (1978), 1-10.
doi: 10.1016/0025-5564(78)90025-1. |
[32] |
Y. Huang and O. Diekmann,
Predator migration in response to prey density: What are the consequences?, Journal of Mathematical Biology, 43 (2001), 561-581.
doi: 10.1007/s002850100107. |
[33] |
V. Hutson,
A theorem on average liapunov functions, Monatshefte für Mathematik, 98 (1984), 267-275.
doi: 10.1007/BF01540776. |
[34] |
V. Hutson and K. Schmit,
Permanence and the dynamics of biological systems, Mathematical Biosciences, 111 (1992), 1-71.
doi: 10.1016/0025-5564(92)90078-B. |
[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.
doi: 10.2307/3545983. |
[36] |
V. A. Jansen,
The dynamics of two diffusively coupled predator-prey populations, Theoretical Population Biology, 59 (2001), 119-131.
doi: 10.1006/tpbi.2000.1506. |
[37] |
V. A. A. Jansen,
Theoretical Aspects of Metapopulation Dynamics, PhD thesis, Ph. D. thesis, Leiden University, The Netherlands, 1994. |
[38] |
Y. Kang and D. Armbruster,
Dispersal effects on a discrete two-patch model for plant-insect interactions, Journal of Theoretical Biology, 268 (2011), 84-97.
doi: 10.1016/j.jtbi.2010.09.033. |
[39] |
Y. Kang and C. Castillo-Chavez,
Multiscale analysis of compartment models with dispersal, Journal of Biological Dynamics, 6 (2012), 50-79.
doi: 10.1080/17513758.2012.713125. |
[40] |
P. Kareiva and 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 and 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.
doi: 10.1098/rstb.1990.0191. |
[42] |
S. Kéfi, M. Rietkerk, M. van Baalen and M. Loreau,
Local facilitation, bistability and transitions in arid ecosystems, Theoretical Population Biology, 71 (2007), 367-379.
|
[43] |
P. Klepac, M. G. Neubert and P. van den Driessche,
Dispersal delays, predator-prey stability, and the paradox of enrichment, Theoretical Population Biology, 71 (2007), 436-444.
doi: 10.1016/j.tpb.2007.02.002. |
[44] |
M. Kummel, D. Brown and A. Bruder,
How the aphids got their spots: Predation drives self-organization of aphid colonies in a patchy habitat, Oikos, 122 (2013), 896-906.
doi: 10.1111/j.1600-0706.2012.20805.x. |
[45] |
K. Kuto and Y. Yamada,
Multiple coexistence states for a prey-predator system with cross-diffusion, Journal of Differential Equations, 197 (2004), 315-348.
doi: 10.1016/j.jde.2003.08.003. |
[46] |
I. Lengyel and 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.
doi: 10.1063/1.165819. |
[47] |
S. A. Levin,
Dispersion and population interactions, American Naturalist, 108 (1974), 207-228.
doi: 10.1086/282900. |
[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.
doi: 10.1093/besa/15.3.237. |
[49] |
Z.-z. Li, M. Gao, C. Hui, X.-z. Han and H. Shi,
Impact of predator pursuit and prey evasion on synchrony and spatial patterns in metapopulation, Ecological Modelling, 185 (2005), 245-254.
doi: 10.1016/j.ecolmodel.2004.12.008. |
[50] |
X. Liu and 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.
doi: 10.1016/S0960-0779(02)00408-3. |
[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 and M. E. Byers,
Role of feeding-induced plant volatiles in aggregative behavior of the japanese beetle (coleoptera: Scarabaeidae), Environmental Entomology, 25 (1996), 1188-1191.
doi: 10.1093/ee/25.5.1188. |
[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.
doi: 10.1017/S0007485300011214. |
[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.
doi: 10.1515/9781400882175-003. |
[55] |
R. M. May,
Host-parasitoid systems in patchy environments: A phenomenological model, The Journal of Animal Ecology, 47 (1978), 833-844.
doi: 10.2307/3674. |
[56] |
R. McMurtrie,
Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments, Mathematical Biosciences, 39 (1978), 11-51.
doi: 10.1016/0025-5564(78)90026-3. |
[57] |
T. F. Miller, D. J. Mladenoff and 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 and A. Stewart-Oaten,
Aggregation and stability in metapopulation models, American Naturalist, 140 (1992), 41-58.
doi: 10.1086/285402. |
[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.
doi: 10.1098/rspb.1993.0001. |
[60] |
M. Rees, P. J. Grubb and 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.
doi: 10.1086/285837. |
[61] |
M. Rietkerk and J. Van de Koppel,
Regular pattern formation in real ecosystems, Trends in Ecology & Evolution, 23 (2008), 169-175.
doi: 10.1016/j.tree.2007.10.013. |
[62] |
P. Rohani and G. D. Ruxton,
Dispersal and stability in metapopulations, Mathematical Medicine and Biology, 16 (1999), 297-306.
doi: 10.1093/imammb/16.3.297. |
[63] |
M. L. Rosenzweig and R. H. MacArthur,
Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223.
doi: 10.1086/282272. |
[64] |
G. D. Ruxton,
Density-dependent migration and stability in a system of linked populations, Bulletin of Mathematical Biology, 58 (1996), 643-660.
doi: 10.1007/BF02459477. |
[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.
doi: 10.1007/978-1-4613-4274-8_42. |
[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 and O. Reichman,
Spatial signature of environmental heterogeneity, dispersal, and competition in successional grasslands, Ecological Monographs, 75 (2005), 199-214.
doi: 10.1890/03-0841. |
[69] |
G. Seifert and L. Markus,
Contributions to the Theory of Nonlinear Oscillations ,Princeton University Press, 1956. |
[70] |
Y. Shahak, E. Gal, Y. Offir and 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.
doi: 10.17660/ActaHortic.2008.797.8. |
[71] |
R. V. Solé and J. Bascompte,
Self-Organization in Complex Ecosystems ,Princeton University Press, Princeton, 2006. |
[72] |
A. Soro, S. Sundberg and H. Rydin,
Species diversity, niche metrics and species associations in harvested and undisturbed bogs, Journal of Vegetation Science, 10 (1999), 549-560.
doi: 10.2307/3237189. |
[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 and 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 and J. Wei,
Predator-prey system with strong Allee effect in prey, Journal of Mathematical Biology, 62 (2011), 291-331.
doi: 10.1007/s00285-010-0332-1. |




Scenarios | Model (4) whose dispersal is driven by the strength of prey-predator interactions | Classical Model (12) whose dispersal is driven by the density of predators |
LAS and GAS if |
GAS if |
|
LAS if |
Does not exists | |
Does not exists | LAS if |
Scenarios | Model (4) whose dispersal is driven by the strength of prey-predator interactions | Classical Model (12) whose dispersal is driven by the density of predators |
LAS and GAS if |
GAS if |
|
LAS if |
Does not exists | |
Does not exists | LAS if |
Scenarios | Model (4) whose dispersal is driven by the strength of prey-predator interactions | Classical Model (12) whose dispersal is driven by the density of predators |
Persistence of prey | Always persist, dispersal of predator has no effects | One or both prey persist if conditions 4. in Theorem (4.1) holds. Small dispersal of predator in Patch |
Extinction of prey | Never extinct |
Scenarios | Model (4) whose dispersal is driven by the strength of prey-predator interactions | Classical Model (12) whose dispersal is driven by the density of predators |
Persistence of prey | Always persist, dispersal of predator has no effects | One or both prey persist if conditions 4. in Theorem (4.1) holds. Small dispersal of predator in Patch |
Extinction of prey | Never extinct |
Scenarios | Model (4) whose dispersal is driven by the strength of prey-predator interactions | Classical Model (12) whose dispersal is driven by the density of predators |
Persistence of predator | Predator at Patch |
Predators in both patches have the same persistence conditions. They persist if |
Extinction of predator | Simulations suggestions (see the yellow regions of Figure (1a) and Figure (3a)) that the large dispersal of predator in Patch |
Predators in both patches have the same extinction conditions. They go extinct if |
Scenarios | Model (4) whose dispersal is driven by the strength of prey-predator interactions | Classical Model (12) whose dispersal is driven by the density of predators |
Persistence of predator | Predator at Patch |
Predators in both patches have the same persistence conditions. They persist if |
Extinction of predator | Simulations suggestions (see the yellow regions of Figure (1a) and Figure (3a)) that the large dispersal of predator in Patch |
Predators in both patches have the same extinction conditions. They go extinct if |
[1] |
Jinfeng Wang, Hongxia Fan. Dynamics in a Rosenzweig-Macarthur predator-prey system with quiescence. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 909-918. doi: 10.3934/dcdsb.2016.21.909 |
[2] |
Xiaoqing Lin, Yancong Xu, Daozhou Gao, Guihong Fan. Bifurcation and overexploitation in Rosenzweig-MacArthur model. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022094 |
[3] |
Razvan C. Fetecau, Beril Zhang. Self-organization on Riemannian manifolds. Journal of Geometric Mechanics, 2019, 11 (3) : 397-426. doi: 10.3934/jgm.2019020 |
[4] |
Komi Messan, Yun Kang. A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 947-976. doi: 10.3934/dcdsb.2017048 |
[5] |
Xun Cao, Xianyong Chen, Weihua Jiang. Bogdanov-Takens bifurcation with $ Z_2 $ symmetry and spatiotemporal dynamics in diffusive Rosenzweig-MacArthur model involving nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3747-3785. doi: 10.3934/dcds.2022031 |
[6] |
Wei Feng, Nicole Rocco, Michael Freeze, Xin Lu. Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1215-1230. doi: 10.3934/dcdss.2014.7.1215 |
[7] |
Yaying Dong, Shanbing Li, Yanling Li. Effects of dispersal for a predator-prey model in a heterogeneous environment. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2511-2528. doi: 10.3934/cpaa.2019114 |
[8] |
Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737 |
[9] |
R. P. Gupta, Peeyush Chandra, Malay Banerjee. Dynamical complexity of a prey-predator model with nonlinear predator harvesting. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 423-443. doi: 10.3934/dcdsb.2015.20.423 |
[10] |
Malay Banerjee, Nayana Mukherjee, Vitaly Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamics. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2109-2120. doi: 10.3934/dcdss.2020180 |
[11] |
Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244 |
[12] |
Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 |
[13] |
Xinfu Chen, Yuanwei Qi, Mingxin Wang. Steady states of a strongly coupled prey-predator model. Conference Publications, 2005, 2005 (Special) : 173-180. doi: 10.3934/proc.2005.2005.173 |
[14] |
Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536 |
[15] |
Mingxin Wang, Peter Y. H. Pang. Qualitative analysis of a diffusive variable-territory prey-predator model. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1061-1072. doi: 10.3934/dcds.2009.23.1061 |
[16] |
Yu-Xia Hao, Wan-Tong Li, Fei-Ying Yang. Traveling waves in a nonlocal dispersal predator-prey model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3113-3139. doi: 10.3934/dcdss.2020340 |
[17] |
J. Gani, R. J. Swift. Prey-predator models with infected prey and predators. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5059-5066. doi: 10.3934/dcds.2013.33.5059 |
[18] |
Siyu Liu, Haomin Huang, Mingxin Wang. A free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1649-1670. doi: 10.3934/dcdsb.2019245 |
[19] |
Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172 |
[20] |
Pankaj Kumar, Shiv Raj. Modelling and analysis of prey-predator model involving predation of mature prey using delay differential equations. Numerical Algebra, Control and Optimization, 2021 doi: 10.3934/naco.2021035 |
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]