May  2017, 22(3): 947-976. doi: 10.3934/dcdsb.2017048

A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects

1. 

Simon A. Levin Mathematical and Computational Modeling Sciences Center, Arizona State University, Tempe, AZ 85287, USA

2. 

Sciences and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA

* Corresponding author: Yun Kang

Received  September 2015 Revised  April 2016 Published  January 2017

Fund Project: K.S.M is partially supported by the Department of Education GAANN (P200A120192). This research of Y.K. 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)

We propose and study a two patch Rosenzweig-MacArthur prey-predator model with immobile prey and predator using two dispersal strategies. The first dispersal strategy is driven by the prey-predator interaction strength, and the second dispersal is prompted by the local population density of predators which is referred as the passive dispersal. The dispersal strategies using by predator are measured by the proportion of the predator population using the passive dispersal strategy which is a parameter ranging from 0 to 1. We focus on how the dispersal strategies and the related dispersal strengths affect population dynamics of prey and predator, hence generate different spatial dynamical patterns in heterogeneous environment. We provide local and global dynamics of the proposed model. Based on our analytical and numerical analysis, interesting findings could be summarized as follow: (1) If there is no prey in one patch, then the large value of dispersal strength and the large predator population using the passive dispersal in the other patch could drive predator extinct at least locally. However, the intermediate predator population using the passive dispersal could lead to multiple interior equilibria and potentially stabilize the dynamics; (2) The large dispersal strength in one patch may stabilize the boundary equilibrium and lead to the extinction of predator in two patches locally when predators use two dispersal strategies; (3) For symmetric patches (i.e., all the life history parameters are the same except the dispersal strengths), the large predator population using the passive dispersal can generate multiple interior attractors; (4) The dispersal strategies can stabilize the system, or destabilize the system through generating multiple interior equilibria that lead to multiple attractors; and (5) The large predator population using the passive dispersal could lead to no interior equilibrium but both prey and predator can coexist through fluctuating dynamics for almost all initial conditions.

Citation: Komi Messan, Yun Kang. A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 947-976. doi: 10.3934/dcdsb.2017048
References:
[1]

P. Auger and J. Poggiale, Emergence of population growth models: Fast migration and slow growth, Journal of Theoretical Biology, 182 (1996), 99-108. doi: 10.1006/jtbi.1996.0145. Google Scholar

[2]

D. E. Bowler and T. G. Benton, Causes and consequences of animal dispersal strategies: Relating individual behaviour to spatial dynamics, Biological Reviews, 80 (2005), 205-225. doi: 10.1017/S1464793104006645. Google Scholar

[3]

J. Colbert, E. Danchin, A. A. Dhondt and J. D. Nichols, Dispersal Oxford University Press, New York, (2001).Google Scholar

[4]

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. Google Scholar

[5]

G. K. DavorenW. A. Montevecchi and J. T. Anderson, Distributional patterns of a marine bird and its prey: Habitat selection based on prey and conspecific behaviour, Marine Ecology Progress Series, 256 (2003), 229-242. doi: 10.3354/meps256229. Google Scholar

[6]

D. F. Fraser and R. D. Cerri, Experimental evaluation of predator-prey relationships in a patchy environment: consequences for habitat use patterns in minnows, Ecology, 62 (1982), 307-313. doi: 10.2307/1938947. Google Scholar

[7]

J. F. Fujita, M. S. and R. V. Bowen, Ant foraging behavior: Ambient temperature influences prey selection, Behavioral Ecology and Sociobiology, 15 65–68.Google Scholar

[8]

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

[9]

D. Grünbaum and V. R. Richard, Black-browed albatrosses foraging on antarctic krill: density-dependence through local enhancement?, Ecology, 84 (2003), 3265-3275. Google Scholar

[10]

I. Hanski, Metapopulation dynamics, Nature, 396 (1998), 41-49. Google Scholar

[11]

I. Hanski, Habitat connectivity, habitat continuity, and metapopulations in dynamic landscapes, Oikos, 87 (1999), 209-219. doi: 10.2307/3546736. Google Scholar

[12]

I. Hanski, Metapopulation Ecology Oxford University Press, Oxford, 1999.Google Scholar

[13]

I. Hanski and M. Gilpin, Metapopulation dynamics: Brief history and conceptual domain, Biological Journal of the Linnean Society, 42 (1991), 3-16. doi: 10.1016/B978-0-12-284120-0.50004-8. Google Scholar

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L. Hansson, Dispersal and connectivity in metapopulations, Biological Journal of the Linnean Society, 42 (1991), 89-103. doi: 10.1016/B978-0-12-284120-0.50009-7. Google Scholar

[15]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theoretical Population Biology, 24 (1983), 244-251. doi: 10.1016/0040-5809(83)90027-8. Google Scholar

[16]

S. HsuS. Hubbell and P. Wlatman, 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

[17]

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

[18]

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

[19]

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

[20]

R. A. Ims and D. Hjermann, Condition-dependent dispersal, in Dispersal, Oxford University Press, Oxford, (2001), 203–216.Google Scholar

[21]

I. M. J{á}nosi and I. Scheuring, On the evolution of density dependent dispersal in a spatially structured population model, Journal of Theoretical Biology, 187 (1997), 397-408. Google Scholar

[22]

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

[23]

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. Google Scholar

[24]

Y. Kang, S. K. Sasmal and K. Messan, A two-patch prey-predator model with dispersal in predators driven by the strength of predation, preprint, arXiv: 1505.03820.Google Scholar

[25]

P. Kareiva and G. Odell, Swarms of predators exhibit ''prey-taxis" if individual predators use area-restricted search, American Naturalist,, 130 (1987), 233-270. Google Scholar

[26]

A. Kiester and M. Slatkin, A strategy of movement and resource utilization, Theoretical Population Biology, 6 (1974), 1-20. doi: 10.1016/0040-5809(74)90028-8. Google Scholar

[27]

V. Krivan, Dispersal dynamics: Distribution of lady beetles (coleoptera: Coccinellidae), European Journal of Entomology, 105 (2008), 405-409. doi: 10.14411/eje.2008.051. Google Scholar

[28]

M. KummelD. 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. Google Scholar

[29]

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. Google Scholar

[30]

Y. Liu, The Dynamical Behavior of a two Patch Predator-prey Model Honor Thesis, from The College of William and Mary, 2010.Google Scholar

[31]

G. P. Markin, Foraging behavior of the argentine ant in a california citrus grove, Journal of Economic Entomology, 63 (1970), 740-744. doi: 10.1093/jee/63.3.740. Google Scholar

[32]

M. MassotJ. ClobertP. Lorenzon and J.-M. Rossi, Condition-dependent dispersal and ontogeny of the dispersal behaviour: An experimental approach, Journal of Animal Ecology, 71 (2002), 253-261. doi: 10.1046/j.1365-2656.2002.00592.x. Google Scholar

[33]

E. Matthysen, Density-dependent dispersal in birds and mammals, Ecography, 28 (2005), 403-416. doi: 10.1111/j.0906-7590.2005.04073.x. Google Scholar

[34]

T. Namba, Density-dependent dispersal and spatial distribution of a population, Journal of Theoretical Biology, 86 (1980), 351-363. doi: 10.1016/0022-5193(80)90011-9. Google Scholar

[35]

D. Nguyen-NgocT. Nguyen-Huu and P. Auger, Effects of fast density dependent dispersal on pre-emptive competition dynamics, Ecological Complexity, 10 (2012), 26-33. doi: 10.1016/j.ecocom.2011.12.003. Google Scholar

[36]

J. Poggiale, From behavioural to population level: Growth and competition, Mathematical and Computer Modelling, 27 (1998), 41-49. doi: 10.1016/S0895-7177(98)00004-1. Google Scholar

[37]

J. D. Reeve, Environmental variability, migration, and persistence in host-parasitoid systems, American Naturalist, 132 (1988), 810-836. doi: 10.1086/284891. Google Scholar

[38] O. RonceI. OlivieriJ. Clobert and E. G. Danchin, Perspectives on the study of dispersal evolution, Oxford University Press, Oxford, 2001. Google Scholar
[39]

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. Google Scholar

[40]

J. F. Savino and R. A. Stein, Behavioural interactions between fish predators and their prey: effects of plant density, Animal Behaviour, 37 (1989), 311-321. doi: 10.1016/0003-3472(89)90120-6. Google Scholar

[41]

J. A. SilvaM. L. De Castro and D. A. Justo, Stability in a metapopulation model with density-dependent dispersal, Bulletin of Mathematical Biology, 63 (2001), 485-505. doi: 10.1006/bulm.2000.0221. Google Scholar

[42]

A. T. Smith and M. M. Peacock, Conspecific attraction and the determination of metapopulation colonization rates, Conservation Biology, 4 (1990), 320-323. doi: 10.1111/j.1523-1739.1990.tb00294.x. Google Scholar

[43]

J. Stamps, Conspecific attraction and aggregation in territorial species, American Naturalist, 131 (1988), 329-347. doi: 10.1086/284793. Google Scholar

[44]

Foraging behavior of ants: experiments with two species of myrmecine ants, Behavioral Ecology and Sociobiology, 2 (1977), 147–167.Google Scholar

[45]

H. R. Thieme, Mathematics in Population Biology Princeton University Press, 2003.Google Scholar

show all references

References:
[1]

P. Auger and J. Poggiale, Emergence of population growth models: Fast migration and slow growth, Journal of Theoretical Biology, 182 (1996), 99-108. doi: 10.1006/jtbi.1996.0145. Google Scholar

[2]

D. E. Bowler and T. G. Benton, Causes and consequences of animal dispersal strategies: Relating individual behaviour to spatial dynamics, Biological Reviews, 80 (2005), 205-225. doi: 10.1017/S1464793104006645. Google Scholar

[3]

J. Colbert, E. Danchin, A. A. Dhondt and J. D. Nichols, Dispersal Oxford University Press, New York, (2001).Google Scholar

[4]

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. Google Scholar

[5]

G. K. DavorenW. A. Montevecchi and J. T. Anderson, Distributional patterns of a marine bird and its prey: Habitat selection based on prey and conspecific behaviour, Marine Ecology Progress Series, 256 (2003), 229-242. doi: 10.3354/meps256229. Google Scholar

[6]

D. F. Fraser and R. D. Cerri, Experimental evaluation of predator-prey relationships in a patchy environment: consequences for habitat use patterns in minnows, Ecology, 62 (1982), 307-313. doi: 10.2307/1938947. Google Scholar

[7]

J. F. Fujita, M. S. and R. V. Bowen, Ant foraging behavior: Ambient temperature influences prey selection, Behavioral Ecology and Sociobiology, 15 65–68.Google Scholar

[8]

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

[9]

D. Grünbaum and V. R. Richard, Black-browed albatrosses foraging on antarctic krill: density-dependence through local enhancement?, Ecology, 84 (2003), 3265-3275. Google Scholar

[10]

I. Hanski, Metapopulation dynamics, Nature, 396 (1998), 41-49. Google Scholar

[11]

I. Hanski, Habitat connectivity, habitat continuity, and metapopulations in dynamic landscapes, Oikos, 87 (1999), 209-219. doi: 10.2307/3546736. Google Scholar

[12]

I. Hanski, Metapopulation Ecology Oxford University Press, Oxford, 1999.Google Scholar

[13]

I. Hanski and M. Gilpin, Metapopulation dynamics: Brief history and conceptual domain, Biological Journal of the Linnean Society, 42 (1991), 3-16. doi: 10.1016/B978-0-12-284120-0.50004-8. Google Scholar

[14]

L. Hansson, Dispersal and connectivity in metapopulations, Biological Journal of the Linnean Society, 42 (1991), 89-103. doi: 10.1016/B978-0-12-284120-0.50009-7. Google Scholar

[15]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theoretical Population Biology, 24 (1983), 244-251. doi: 10.1016/0040-5809(83)90027-8. Google Scholar

[16]

S. HsuS. Hubbell and P. Wlatman, 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

[17]

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

[18]

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

[19]

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

[20]

R. A. Ims and D. Hjermann, Condition-dependent dispersal, in Dispersal, Oxford University Press, Oxford, (2001), 203–216.Google Scholar

[21]

I. M. J{á}nosi and I. Scheuring, On the evolution of density dependent dispersal in a spatially structured population model, Journal of Theoretical Biology, 187 (1997), 397-408. Google Scholar

[22]

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

[23]

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. Google Scholar

[24]

Y. Kang, S. K. Sasmal and K. Messan, A two-patch prey-predator model with dispersal in predators driven by the strength of predation, preprint, arXiv: 1505.03820.Google Scholar

[25]

P. Kareiva and G. Odell, Swarms of predators exhibit ''prey-taxis" if individual predators use area-restricted search, American Naturalist,, 130 (1987), 233-270. Google Scholar

[26]

A. Kiester and M. Slatkin, A strategy of movement and resource utilization, Theoretical Population Biology, 6 (1974), 1-20. doi: 10.1016/0040-5809(74)90028-8. Google Scholar

[27]

V. Krivan, Dispersal dynamics: Distribution of lady beetles (coleoptera: Coccinellidae), European Journal of Entomology, 105 (2008), 405-409. doi: 10.14411/eje.2008.051. Google Scholar

[28]

M. KummelD. 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. Google Scholar

[29]

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. Google Scholar

[30]

Y. Liu, The Dynamical Behavior of a two Patch Predator-prey Model Honor Thesis, from The College of William and Mary, 2010.Google Scholar

[31]

G. P. Markin, Foraging behavior of the argentine ant in a california citrus grove, Journal of Economic Entomology, 63 (1970), 740-744. doi: 10.1093/jee/63.3.740. Google Scholar

[32]

M. MassotJ. ClobertP. Lorenzon and J.-M. Rossi, Condition-dependent dispersal and ontogeny of the dispersal behaviour: An experimental approach, Journal of Animal Ecology, 71 (2002), 253-261. doi: 10.1046/j.1365-2656.2002.00592.x. Google Scholar

[33]

E. Matthysen, Density-dependent dispersal in birds and mammals, Ecography, 28 (2005), 403-416. doi: 10.1111/j.0906-7590.2005.04073.x. Google Scholar

[34]

T. Namba, Density-dependent dispersal and spatial distribution of a population, Journal of Theoretical Biology, 86 (1980), 351-363. doi: 10.1016/0022-5193(80)90011-9. Google Scholar

[35]

D. Nguyen-NgocT. Nguyen-Huu and P. Auger, Effects of fast density dependent dispersal on pre-emptive competition dynamics, Ecological Complexity, 10 (2012), 26-33. doi: 10.1016/j.ecocom.2011.12.003. Google Scholar

[36]

J. Poggiale, From behavioural to population level: Growth and competition, Mathematical and Computer Modelling, 27 (1998), 41-49. doi: 10.1016/S0895-7177(98)00004-1. Google Scholar

[37]

J. D. Reeve, Environmental variability, migration, and persistence in host-parasitoid systems, American Naturalist, 132 (1988), 810-836. doi: 10.1086/284891. Google Scholar

[38] O. RonceI. OlivieriJ. Clobert and E. G. Danchin, Perspectives on the study of dispersal evolution, Oxford University Press, Oxford, 2001. Google Scholar
[39]

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. Google Scholar

[40]

J. F. Savino and R. A. Stein, Behavioural interactions between fish predators and their prey: effects of plant density, Animal Behaviour, 37 (1989), 311-321. doi: 10.1016/0003-3472(89)90120-6. Google Scholar

[41]

J. A. SilvaM. L. De Castro and D. A. Justo, Stability in a metapopulation model with density-dependent dispersal, Bulletin of Mathematical Biology, 63 (2001), 485-505. doi: 10.1006/bulm.2000.0221. Google Scholar

[42]

A. T. Smith and M. M. Peacock, Conspecific attraction and the determination of metapopulation colonization rates, Conservation Biology, 4 (1990), 320-323. doi: 10.1111/j.1523-1739.1990.tb00294.x. Google Scholar

[43]

J. Stamps, Conspecific attraction and aggregation in territorial species, American Naturalist, 131 (1988), 329-347. doi: 10.1086/284793. Google Scholar

[44]

Foraging behavior of ants: experiments with two species of myrmecine ants, Behavioral Ecology and Sociobiology, 2 (1977), 147–167.Google Scholar

[45]

H. R. Thieme, Mathematics in Population Biology Princeton University Press, 2003.Google Scholar

Figure 1.  One parameter bifurcation diagrams of Model (4) with $y$-axis representing the population size of predator at Patch 1 and $x$-axis represent the proportion of predator using the passive dispersal. Figure 1(a) describes the number of interior equilibria $(\hat{x}_1^*, \hat{y}_1^*, \hat{y}_2^*)$ when $x_2 = 0$ in Model (3) and their stability with respect to variation in $s$. Figure 1(b) and 1(c) describe the number of interior equilibria $(y_1^*, x_2^*, y_2^*)$ of the submodel $x_2 = 0$ of Model (3) and their stability when $s$ varies from $0$ to $1$. Blue represents the sink and green represents the saddle.
Figure 2.  Boundary equilibria $E_{1\ell}^b = (x_{1\ell}^*, y_{1\ell}^*, 0, y_{2\ell}^*)$ and $E_{2\ell}^b = (0, \hat{y}_{1\ell}^*, \hat{x}_{2\ell}^*, \hat{y}_{2\ell}^*)$. Figure 2(a) and 2(b) are the cases when $s = 0.65$, $r_1=1$, $r_2=0.54$, $d_1=0.45$, $d_2=0.105$, $K_1=10$, $K_2=8$, $a_1=0.6$, $a_2=0.35$, $\rho_1 = 1.75$, and $\rho_2=1.2$. The solid lines are $f_1(x_1)$ and $f_2(x_2)$ while the dashed lines are $K_1$ and $K_2$ which illustrates the existence of boundary equilibria when $K_1> x_{1\ell}^* \mbox{ or } K_2> \hat{x}_{2\ell}^*, ~\ell=1, 2$. The black dots represent real positive $x_{1\ell}^*$ and $\hat{x}_{2\ell}^*$ that satisfy existence of boundary equilibria, respectively.
Figure 3.  One parameter bifurcation diagrams of Model (3) with $y$-axis representing the population size of predator at Patch 1 and $x$-axis represent the proportion of predator using the passive dispersal. Figure 3(a) describes the number of boundary equilibria $E_{1\ell}^b = (x_{1\ell}^*, y_{1\ell}^*, 0, y_{2\ell}^*), \ell=1, 2$ from Model (3) and their stability with respect to variation in $s$ when $d_1 = 0.85, ~a_1=1$. Figure 3(b) and 3(c) describes the number of boundary equilibria $E_{2\ell}^b = (0, \hat{y}_{1\ell}^*, \hat{x}_{2\ell}^*, \hat{y}_{2\ell}^*), \ell =1, 2$ from Model (3) and their change in stability when $s$ varies from $0$ to $1$ with $d_1 = 0.85, ~a_1=1$ and $d_1 = 2, ~a_1=2.1$ respectively. Blue represents the sink and green represents the saddle.
Figure 4.  One and two parameter bifurcation diagrams of symmetric Model (3) with $y$-axis representing the population size of predator at Patch 1 in figure 4(a). We used the following parameters $r=1$, $d=5$, $K=10$, and $a=6$. Figure 4(a) describes the number of interior equilibria and their change in stability when $s$ varies from $0$ to $1$. Blue line represents sink and green line represents saddle in Figure 4(a). Figure 4(b) describes how the number of interior equilibria change for different values of dispersal strategy $s$ and dispersal rate $\rho_1$. Black region have three interior equilibria; red regions have two interior equilibria; and blue regions have one interior equilibrium in Figure 4(b)
Figure 5.  One and two parameter bifurcation diagrams of Model (3) with $y$-axis representing the population size of predator at Patch 1 in Figure 5(a). The following parameters are used: $r=1.8$, $d_2=0.35$, $K_1=10$, $K_2=7$, and $a_2=1.4$. Figure 5(a) describes the number of interior equilibria and their change in stability when $s$ varies from $0$ to $1$. Blue line represents sink, green line represents saddle, and red line represents source in Figure 5(a). Figure 5(b) describes how the number of interior equilibria change for different values of $s$ and dispersal rate $\rho_1$. Black region have three interior equilibria; red regions have two interior equilibria; blue regions have one interior equilibrium, and white regions have no interior equilibria in Figure 5(b).
Figure 6.  One and two parameter bifurcation diagrams of Model (3) with $y$-axis representing the population size of predator at Patch 1 in figure 6(a). We used the following parameters $r=1.8$, $d_2=0.35$, $K_1=10$, $K_2=7$, and $a_2=1.4$. Figure 6(a) describes the number of interior equilibria and their change in stability when $s$ varies from $0$ to $1$ under the parameters $d_1 =2$ and $a_1=2.1$. Blue represents sink, green represents saddle, and red represents source. Figure 6(b) describes how the number of interior equilibria change for different values of dispersal strategy $s$ and dispersal rate $\rho_1$. Black region have three interior equilibria; red regions have two interior equilibria; blue regions have one interior equilibrium, and white regions have no interior equilibria
Figure 7.  Two parameters bifurcation diagrams of Model (3) with $y$-axis representing the relative dispersal rate $\rho_2$ and $x$-axis represent the strength of dispersal mode $s$. The following parameters are used: $r=1.8$, $d_2=0.35$, $K_1=10$, $K_2=7$, $a_2=1.4$, and $\rho_1 = 1$. Both figures 7(a) and 7(b) describes how the number of interior equilibria change for different values of dispersal strategy $s$ and dispersal rate $\rho_2$ where the parameters $d_1 =0.85, ~a_1=1$ are used for the left figure 7(a) while $d_1 =2, ~a_1=2.1$ are used for the right figure 7(b) in addition to the fixed parameters. Black region have three interior equilibria; red regions have two interior equilibria; blue regions have one interior equilibrium, and white regions have no interior equilibria in Figures 7(a) and 7(b)
Figure 8.  Time series of Model 3 when $r=1.8$, $d_2=0.35$, $K_1=10$, $K_2=7$, and $a_2=1.4$. Figures 8(a) and 8(b) illustrate the coexistence of prey and predator through fluctuating dynamics while Model 3 has no interior equilibria. The blue dashed lines represent the prey population in patch 1, the dashed red lines represent the predator population in patch 1, the blue solid lines is the the prey population in patch 2, and the red solid lines represent predator population in patch 2
Figure 9.  Time series of Model 3 when $r=1.8$, $d_2=0.35$, $K_1=10$, $K_2=7$, $a_2=1.4$, $\rho_1 = 1$, and $\rho_2 = 2.5$. Figures 9(a) and 9(b) represent the dynamical pattern generated by two interior saddles, one boundary sink and one boundary saddle. The blue dashed lines represent the prey population in patch 1, the dashed red lines represent the predator population in patch 1, the blue solid lines is the the prey population in patch 2, and the red solid lines represent predator population in patch 2.
Figure 10.  Time series of Model 3 when $r=1.8$, $d_2=0.35$, $K_1=10$, $K_2=7$, $a_2=1.4$, $\rho_1 = 1$, and $\rho_2 = 2.5$. Figures 10(a) and 10(b) describe the dynamical pattern generated by two interior saddles and one interior that is locally stable. The blue dashed lines represent the prey population in patch 1, the dashed red lines represent the predator population in patch 1, the blue solid lines is the the prey population in patch 2, and the red solid lines represent predator population in patch 2
Table 1.  Summary of the effect of the proportion of predators using the passive dispersal on Model (4) From Figures 1(a), 1(b), and 1(c). LAS refers to local asymptotical stability and ✗ implies the equilibrium does not exist
Scenarios $\mathbf{a_1= 1 \mbox{ and } d_1 =0.85 }$ $\mathbf{a_1= 2.1 \mbox{ and } d_1 = 2 }$
$\mathbf{E_{x_1y_1y_2}^{1} }$ $\mathbf{E_{x_1y_1y_2}^{2} }$ $\mathbf{ E_{y_1x_2y_2}^{1}}$ $\mathbf{E_{y_1x_2y_2}^{2} }$ $\mathbf{E_{x_1y_1y_2}^{1, 2} }$ $\mathbf{ E_{y_1x_2y_2}^{1}}$ $\mathbf{E_{y_1x_2y_2}^{2} }$
$s\leq0.1$ LAS Saddle Saddle
$ 0.15 \leq s \leq 0.45$ LAS Saddle Saddle Saddle
$ 0.55 \leq s \leq 0.62$ Saddle LAS Saddle
$ 0.68 < s < 0.82$ LAS Saddle
$ s \geq 0.82$
Scenarios $\mathbf{a_1= 1 \mbox{ and } d_1 =0.85 }$ $\mathbf{a_1= 2.1 \mbox{ and } d_1 = 2 }$
$\mathbf{E_{x_1y_1y_2}^{1} }$ $\mathbf{E_{x_1y_1y_2}^{2} }$ $\mathbf{ E_{y_1x_2y_2}^{1}}$ $\mathbf{E_{y_1x_2y_2}^{2} }$ $\mathbf{E_{x_1y_1y_2}^{1, 2} }$ $\mathbf{ E_{y_1x_2y_2}^{1}}$ $\mathbf{E_{y_1x_2y_2}^{2} }$
$s\leq0.1$ LAS Saddle Saddle
$ 0.15 \leq s \leq 0.45$ LAS Saddle Saddle Saddle
$ 0.55 \leq s \leq 0.62$ Saddle LAS Saddle
$ 0.68 < s < 0.82$ LAS Saddle
$ s \geq 0.82$
Table 2.  Summary of the effect of the proportion of predators using the passive dispersal on Model (4) From Figures 3(a), 3(b), and 3(c). LAS refers to local asymptotical stability and ✗ implies the equilibrium does not exist.
Scenarios $\mathbf{a_1= 1 \mbox{ and } d_1 =0.85 }$ $\mathbf{a_1= 2.1 \mbox{ and } d_1 = 2 }$
E11b E12b E21b E22b E11, 12b E21b E22b
s ≤ 0.1 Saddle Saddle Saddle
0.15 ≤ s ≤ 0.45 Saddle Saddle Saddle Saddle
0.55 ≤ s ≤ 0.62 Saddle Saddle Saddle
0.68 < s < 0.82 LAS Saddle
s ≥ 0.82
Scenarios $\mathbf{a_1= 1 \mbox{ and } d_1 =0.85 }$ $\mathbf{a_1= 2.1 \mbox{ and } d_1 = 2 }$
E11b E12b E21b E22b E11, 12b E21b E22b
s ≤ 0.1 Saddle Saddle Saddle
0.15 ≤ s ≤ 0.45 Saddle Saddle Saddle Saddle
0.55 ≤ s ≤ 0.62 Saddle Saddle Saddle
0.68 < s < 0.82 LAS Saddle
s ≥ 0.82
Table 3.  Summary of the local and global dynamic of Model (3). LAS refers to the local asymptotical stability, GAS refers to the global stability, and Cond. refers to condition.
Scenarios Existence condition, Local and Global stability of Model (3)
$\mathbf{ s = 0}$ $\mathbf{ s \in (0, 1)}$ $\mathbf{ s = 1}$
$E_{0000}, $$E_{K_1000}, $$E_{00K_20}$ Always exist and always saddle Always exist and always saddle Always exist and always saddle
$E_{K_10K_20}$ Always exist; LAS and GAS if $\mu_i>K_i$ for both $i=1, 2$ Always exist; GAS if $\mu_i>K_i$ for both $i=1, 2$; while LAS if Equations 8 are satisfied Always exist; GAS if $\mu_i>K_i$ for both $i=1, 2$; LAS if condition (1) is satisfied
$E_{1\ell}^b$
$x_i=0$
$\ell = 1, ~2$$i =1, 2$
Do not exist One or two exist if $\frac{3\beta_j}{\mu_j+K_j}<\alpha_j<(\mu_j+K_j)^2$ with $i, j=1, 2$, $i\not=j$; Can be locally asymptotically stable or saddle as shown in Figures 3(a), 3(b), 3(c) Exist if $0<\hat{\mu}_i<K_i$; LAS if $\frac{K_i-1}{2}<\widehat{\mu}_i<K_i$ and $r_j<a_j\hat{\nu}_j^i$. GAS if $\frac{K_i-1}{2}<\widehat{\mu}_i<K_i$ and $\frac{r_j(K_j+1)^2}{4a_jK_j}<\widehat{\nu}_i^j$, $i, j=1, 2$, $i\not=j$}.
$E_{i2}^{b*}$
$i, j=1, 2$
$i\neq j$
Exist if $0<\mu_i<K_i$; LAS if $\frac{K_i-1}{2}<\mu_i<K_i$ and condition (2) is satisfied Do not exist Do not exist
Cond. 1:
Cond. 2:
$0<\frac{d_i}{a_j-d_i}<K_j<\mu_j$ and $\rho_j<\frac{d_j-K_j(a_j-d_j)}{\nu_i\left[K_j(a_j-d_i)-d_i\right]}$; $i, j=1, 2$, $i\not=j$
Scenarios Existence condition, Local and Global stability of Model (3)
$\mathbf{ s = 0}$ $\mathbf{ s \in (0, 1)}$ $\mathbf{ s = 1}$
$E_{0000}, $$E_{K_1000}, $$E_{00K_20}$ Always exist and always saddle Always exist and always saddle Always exist and always saddle
$E_{K_10K_20}$ Always exist; LAS and GAS if $\mu_i>K_i$ for both $i=1, 2$ Always exist; GAS if $\mu_i>K_i$ for both $i=1, 2$; while LAS if Equations 8 are satisfied Always exist; GAS if $\mu_i>K_i$ for both $i=1, 2$; LAS if condition (1) is satisfied
$E_{1\ell}^b$
$x_i=0$
$\ell = 1, ~2$$i =1, 2$
Do not exist One or two exist if $\frac{3\beta_j}{\mu_j+K_j}<\alpha_j<(\mu_j+K_j)^2$ with $i, j=1, 2$, $i\not=j$; Can be locally asymptotically stable or saddle as shown in Figures 3(a), 3(b), 3(c) Exist if $0<\hat{\mu}_i<K_i$; 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 $\frac{r_j(K_j+1)^2}{4a_jK_j}<\widehat{\nu}_i^j$, $i, j=1, 2$, $i\not=j$}.
$E_{i2}^{b*}$
$i, j=1, 2$
$i\neq j$
Exist if $0<\mu_i<K_i$; LAS if $\frac{K_i-1}{2}<\mu_i<K_i$ and condition (2) is satisfied Do not exist Do not exist
Cond. 1:
Cond. 2:
$0<\frac{d_i}{a_j-d_i}<K_j<\mu_j$ and $\rho_j<\frac{d_j-K_j(a_j-d_j)}{\nu_i\left[K_j(a_j-d_i)-d_i\right]}$; $i, j=1, 2$, $i\not=j$
Table 4.  Summary of the effect of the proportion of predators using the passive dispersal on the interior equilibria of Model (3) From Figures 5(a), and 6(a). LAS refers to local asymptotical stability, ✗ implies the equilibrium does not exist, and $E_{x_1y_1x_2y_2}^{i}, i=1, 2, 3$ are the three possible interior equilibria of Model (3)
Scenarios $\mathbf{a_1= 1 \mbox{ and } d_1 =0.85 }$ $\mathbf{a_1= 2.1 \mbox{ and } d_1 = 2 }$
${\bf{E}}_{{{\bf{x}}_{\bf{1}}}{{\bf{y}}_{\bf{1}}}{{\bf{x}}_2}{{\bf{y}}_{\bf{2}}}}^{\bf{1}}$ ${\bf{E}}_{{{\bf{x}}_{\bf{1}}}{{\bf{y}}_{\bf{1}}}{{\bf{x}}_2}{{\bf{y}}_{\bf{2}}}}^2$ ${\bf{E}}_{{{\bf{x}}_{\bf{1}}}{{\bf{y}}_{\bf{1}}}{{\bf{x}}_2}{{\bf{y}}_{\bf{2}}}}^3$ ${\bf{E}}_{{{\bf{x}}_{\bf{1}}}{{\bf{y}}_{\bf{1}}}{{\bf{x}}_2}{{\bf{y}}_{\bf{2}}}}^{\bf{1}}$ ${\bf{E}}_{{{\bf{x}}_{\bf{1}}}{{\bf{y}}_{\bf{1}}}{{\bf{x}}_2}{{\bf{y}}_{\bf{2}}}}^2$ ${\bf{E}}_{{{\bf{x}}_{\bf{1}}}{{\bf{y}}_{\bf{1}}}{{\bf{x}}_2}{{\bf{y}}_{\bf{2}}}}^3$
s ≤ 0.07 Source Saddle Source LAS
0.9 ≤ s ≤ 0.15 Source Saddle Saddle LAS
0.2 ≤ s ≤ 0.43 Saddle LAS Saddle Saddle
0.68 < s < 0.82 LAS
s ≥ 0.82
0.83 ≤ s ≤ 0.84 Saddle Saddle LAS Saddle
s ≥ 0.84 Saddle Saddle Saddle Saddle
Scenarios $\mathbf{a_1= 1 \mbox{ and } d_1 =0.85 }$ $\mathbf{a_1= 2.1 \mbox{ and } d_1 = 2 }$
${\bf{E}}_{{{\bf{x}}_{\bf{1}}}{{\bf{y}}_{\bf{1}}}{{\bf{x}}_2}{{\bf{y}}_{\bf{2}}}}^{\bf{1}}$ ${\bf{E}}_{{{\bf{x}}_{\bf{1}}}{{\bf{y}}_{\bf{1}}}{{\bf{x}}_2}{{\bf{y}}_{\bf{2}}}}^2$ ${\bf{E}}_{{{\bf{x}}_{\bf{1}}}{{\bf{y}}_{\bf{1}}}{{\bf{x}}_2}{{\bf{y}}_{\bf{2}}}}^3$ ${\bf{E}}_{{{\bf{x}}_{\bf{1}}}{{\bf{y}}_{\bf{1}}}{{\bf{x}}_2}{{\bf{y}}_{\bf{2}}}}^{\bf{1}}$ ${\bf{E}}_{{{\bf{x}}_{\bf{1}}}{{\bf{y}}_{\bf{1}}}{{\bf{x}}_2}{{\bf{y}}_{\bf{2}}}}^2$ ${\bf{E}}_{{{\bf{x}}_{\bf{1}}}{{\bf{y}}_{\bf{1}}}{{\bf{x}}_2}{{\bf{y}}_{\bf{2}}}}^3$
s ≤ 0.07 Source Saddle Source LAS
0.9 ≤ s ≤ 0.15 Source Saddle Saddle LAS
0.2 ≤ s ≤ 0.43 Saddle LAS Saddle Saddle
0.68 < s < 0.82 LAS
s ≥ 0.82
0.83 ≤ s ≤ 0.84 Saddle Saddle LAS Saddle
s ≥ 0.84 Saddle Saddle Saddle Saddle
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