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August  2017, 14(4): 843-880. doi: 10.3934/mbe.2017046

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

Yun Kang 1,, , Sourav Kumar Sasmal 2, and  Komi Messan 3,

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

* Corresponding author: Yun Kang

Received  August 30, 2016 Accepted  December 25, 2016 Published  February 2017

Fund Project: The first author is partially supported by NSF-DMS(1313312); NSF-IOS/DMS (1558127) and The James S. McDonnell Foundation 21st Century Science Initiative in Studying Complex Systems Scholar Award (UHC Scholar Award 220020472).

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.

Keywords: Rosenzweig-MacArthur prey-predator model, self-organization effects, dispersal, persistence, non-random foraging movements.
Mathematics Subject Classification: Primary: 37G35, 34C23; Secondary: 92D25, 92D40.
Citation: Yun Kang, Sourav Kumar Sasmal, Komi Messan. A two-patch prey-predator model with predator dispersal driven by the predation strength. Mathematical Biosciences & Engineering, 2017, 14 (4) : 843-880. doi: 10.3934/mbe.2017046
References:
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References:
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[14]

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[15]

A. G. Gatehouse, Permanence and the dynamics of biological systems, Host Finding Behaviour Of Tsetse Flies, (1972), 83-95.   Google Scholar

[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.  Google Scholar

[17]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

A. Hastings, Can spatial variation along lead to selection for dispersal?, Theoretical Population Biology, 24 (1983), 244-251.   Google Scholar

[27]

A. Hastings, Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations, Ecology, 74 (1993), 1362-1372.  doi: 10.2307/1940066.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

V. Hutson, A theorem on average liapunov functions, Monatshefte für Mathematik, 98 (1984), 267-275.  doi: 10.1007/BF01540776.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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figure (1b) describes the number of interior equilibria and their stability when $\rho_2=0.025$ and $\rho_1$ changes from 0 to 0.5 where $y$-axis is the population size of predator at Patch 1: Blue represents the sink; green represents the saddle; and red represents the source">Figure 1.  One and two bifurcation diagrams of Model (4) where $r=1.5$, $d_1=0.2$, $d_2=0.1$, $K_1=5$, $K_2=3$, $a_1=0.25$, and $a_2=0.15$. The left figure (1a) describes how number of interior equilibria changes for different dispersal values $\rho_i, i=1,2$: black regions have three interior equilibria; red regions have two interior equilibria; blue regions have unique interior equilibrium; yellow regions have no interior equilibrium and predator in Patch 2 dies out; white regions have no interior equilibrium and both predator die out. The right figure (1b) describes the number of interior equilibria and their stability when $\rho_2=0.025$ and $\rho_1$ changes from 0 to 0.5 where $y$-axis is the population size of predator at Patch 1: Blue represents the sink; green represents the saddle; and red represents the source
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figure (4a) describes describes the number of interior equilibria and their stability when $\rho_1=0.5$ and $\rho_2$ changes from 0 to 0.05. The right figure (4b) describes the number of interior equilibria and their stability when $\rho_1=0.6$ and $\rho_2$ changes from 0 to 1.8. In both figures, blue represents the sink; green represents the saddle; and red represents the source">Figure 4.  One dimensional bifurcation diagrams of Model (4) where $r=1.5$, $d_1=0.2$, $d_2=0.1$, $K_1=5$, $K_2=3$ and $a_1=0.25$. The left figure (4a) describes describes the number of interior equilibria and their stability when $\rho_1=0.5$ and $\rho_2$ changes from 0 to 0.05. The right figure (4b) describes the number of interior equilibria and their stability when $\rho_1=0.6$ and $\rho_2$ changes from 0 to 1.8. In both figures, blue represents the sink; green represents the saddle; and red represents the source
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Figure 2.  One and two bifurcation diagrams of Model (4) where $r=1.5$, $d_1=0.2$, $d_2=0.1$, $K_1=5$, $K_2=3$, $a_1=0.25$ and $a_2=0.25$. The left figure (2a) describes how number of interior equilibria changes for different dispersal values $\rho_i, i=1,2$: black regions have three interior equilibria; red regions have two interior equilibria; blue regions have unique interior equilibrium; yellow regions have no interior equilibrium and predator in Patch 2 dies out; white regions have no interior equilibrium and both predator die out. The right figure (2b) describes the number of interior equilibria and their stability when $\rho_1=1$ and $\rho_2$ changes from 0 to 2.5 where $y$-axis is the population size of predator at Patch 1: Blue represents the sink; green represents the saddle; and red represents the source
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figure (3a) describes how number of interior equilibria changes for different dispersal values $\rho_i, i=1,2$: black regions have three interior equilibria; red regions have two interior equilibria; blue regions have unique interior equilibrium; yellow regions have no interior equilibrium and predator in Patch 2 dies out; white regions have no interior equilibrium and both predator die out. The right figure (3b) describes the number of interior equilibria and their stability when $\rho_1=1$ and $\rho_2$ changes from 0 to 7 where $y$-axis is the population size of predator at Patch 1: Blue represents the sink; green represents the saddle; and red represents the source">Figure 3.  One and two bifurcation diagrams of Model (4) where $r=1.5$, $d_1=0.2$, $d_2=0.1$, $K_1=5$, $K_2=3$, $a_1=0.35$ and $a_2=0.25$. The left figure (3a) describes how number of interior equilibria changes for different dispersal values $\rho_i, i=1,2$: black regions have three interior equilibria; red regions have two interior equilibria; blue regions have unique interior equilibrium; yellow regions have no interior equilibrium and predator in Patch 2 dies out; white regions have no interior equilibrium and both predator die out. The right figure (3b) describes the number of interior equilibria and their stability when $\rho_1=1$ and $\rho_2$ changes from 0 to 7 where $y$-axis is the population size of predator at Patch 1: Blue represents the sink; green represents the saddle; and red represents the source
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Table 1.  The comparison of boundary equilibria between Model (4) and Model (12). LAS refers to the local asymptotical stability, and GAS refers to the global stability
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
$E_{K_10K_20}$ LAS and GAS if $\mu_i>K_i$ for both $i=1,2$. Dispersal has no effects on its stability. GAS if $\mu_i>K_i$ for both $i=1,2$; While LAS if $d_1+d_2+\rho_1+\rho_2>\frac{a_1K_1}{1+K_1}+\frac{a_2K_2}{1+K_2}$ and $\left[ d_1-\frac{a_1K_1}{1+K_1}\right]\left[1-\frac{a_2K_2}{(d_2+\rho_2)(1+K_2)}\right]+\frac{\rho_1}{d_2+\rho_2}\left[ d_2-\frac{a_2K_2}{1+K_2}\right]>0$. Large dispersal may be able to stabilize the equilibrium.
$E_{i2}^b$ ($y_i=0$) LAS if $\frac{K_i-1}{2}<\mu_i<K_i$ and one of the conditions sa, sb, sc, sd in Theorem 3.2 holds. Large dispersal has potential to either stabilize or stabilize the equilibrium. Does not exists
$E_i^b$ ($x_i=0$) Does not exists LAS if $\frac{K_i-1}{2}<\widehat{\mu}_i<K_i$ and $r_j<a_j\hat{\nu}_j^i$. GAS if $\frac{K_i-1}{2}<\widehat{\mu}_i<K_i$ and $\frac{r_j(K_j+1)^2}{4a_jK_j}<\widehat{\nu}_i^j$. Large dispersal of predator in Patch $i$ will either destroy or destabilize the equilibrium while large dispersal of predator in Patch $j$ may stabilize the equilibrium.
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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
$E_{K_10K_20}$ LAS and GAS if $\mu_i>K_i$ for both $i=1,2$. Dispersal has no effects on its stability. GAS if $\mu_i>K_i$ for both $i=1,2$; While LAS if $d_1+d_2+\rho_1+\rho_2>\frac{a_1K_1}{1+K_1}+\frac{a_2K_2}{1+K_2}$ and $\left[ d_1-\frac{a_1K_1}{1+K_1}\right]\left[1-\frac{a_2K_2}{(d_2+\rho_2)(1+K_2)}\right]+\frac{\rho_1}{d_2+\rho_2}\left[ d_2-\frac{a_2K_2}{1+K_2}\right]>0$. Large dispersal may be able to stabilize the equilibrium.
$E_{i2}^b$ ($y_i=0$) LAS if $\frac{K_i-1}{2}<\mu_i<K_i$ and one of the conditions sa, sb, sc, sd in Theorem 3.2 holds. Large dispersal has potential to either stabilize or stabilize the equilibrium. Does not exists
$E_i^b$ ($x_i=0$) Does not exists LAS if $\frac{K_i-1}{2}<\widehat{\mu}_i<K_i$ and $r_j<a_j\hat{\nu}_j^i$. GAS if $\frac{K_i-1}{2}<\widehat{\mu}_i<K_i$ and $\frac{r_j(K_j+1)^2}{4a_jK_j}<\widehat{\nu}_i^j$. Large dispersal of predator in Patch $i$ will either destroy or destabilize the equilibrium while large dispersal of predator in Patch $j$ may stabilize the equilibrium.
Table 2.  The comparison of prey persistence and extinction between Model (4) and Model (12)
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 $i$ and large dispersal of predator in Patch $j$ can help the persistence of prey in Patch $i$.
Extinction of prey Never extinct $x_i$ extinct if $\frac{K_j-1}{2}<\widehat{\mu}_j<K_j$ and $\frac{r_i(K_i+1)^2}{4a_iK_i}<\widehat{\nu}_i^j$. Large dispersal of predator in Patch $i$ can promote the extinction of prey in Patch $i$.
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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 $i$ and large dispersal of predator in Patch $j$ can help the persistence of prey in Patch $i$.
Extinction of prey Never extinct $x_i$ extinct if $\frac{K_j-1}{2}<\widehat{\mu}_j<K_j$ and $\frac{r_i(K_i+1)^2}{4a_iK_i}<\widehat{\nu}_i^j$. Large dispersal of predator in Patch $i$ can promote the extinction of prey in Patch $i$.
Table 3.  The comparison of predator persistence and extinction between Model (4) and Model (12)
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 $j$ is persistent if Conditions in Theorem 3.6 holds. Small dispersal of predator in Patch $j$ can help the persistence of predator in that patch. Dispersal is able to promote the persistence of predator when predator goes extinct in the single patch model. Predators in both patches have the same persistence conditions. They persist if $0<{\mu}_i<K_i$ for $i=1,2$. Dispersal seems to have no effects in the persistence of predator.
Extinction of predator Simulations suggestions (see the yellow regions of Figure (1a) and Figure (3a)) that the large dispersal of predator in Patch $i$ may lead to the its own extinction. Predators in both patches have the same extinction conditions. They go extinct if ${\mu}_i>K_i$ or $\mu_i<0$ for $i=1,2$
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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 $j$ is persistent if Conditions in Theorem 3.6 holds. Small dispersal of predator in Patch $j$ can help the persistence of predator in that patch. Dispersal is able to promote the persistence of predator when predator goes extinct in the single patch model. Predators in both patches have the same persistence conditions. They persist if $0<{\mu}_i<K_i$ for $i=1,2$. Dispersal seems to have no effects in the persistence of predator.
Extinction of predator Simulations suggestions (see the yellow regions of Figure (1a) and Figure (3a)) that the large dispersal of predator in Patch $i$ may lead to the its own extinction. Predators in both patches have the same extinction conditions. They go extinct if ${\mu}_i>K_i$ or $\mu_i<0$ for $i=1,2$
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