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

• * Corresponding author: Yun Kang
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.

Mathematics Subject Classification: Primary:37G35, 34C23;Secondary:92D25, 92D40.

 Citation:

• 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 ✗ ✗ ✗ $s \geq 0.82$ ✗ ✗ ✗ ✗ ✗ ✗ ✗

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 ✗ ✗ ✗ ✗ ✗ ✗ ✗

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 $0<\hat{\mu}_i Table 4. Summary of the effect of the proportion of predators using the passive dispersal on the interior equilibria of Model (3) From Figures 5(a), and 6(a). LAS refers to local asymptotical stability, ✗ implies the equilibrium does not exist, and$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 ✗ ✗
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Tables(4)