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# Pattern formation in a predator-mediated coexistence model with prey-taxis

• Can foraging by predators or a repulsive prey defense mechanism upset predator-mediated coexistence? This paper investigates one scenario involving a prey-taxis by a prey species. We study a system of three populations involving two competing species with a common predator. All three populations are mobile via random dispersal within a bounded spatial domain $\Omega$, but the predator's movement is influenced by one prey's gradient representing a repulsive effect on the predator. We prove existence of positive solutions, and investigate pattern formation through bifurcation analysis and numerical simulation.

Mathematics Subject Classification: 35K59, 35Q92, 92D25.

 Citation: • • Figure 1.  A plot of $\chi$ versus $k^{2}$. The red line is $\chi_{c}$ as a function of $k^{2}$. The blue dashed line shows $a(k^{2})b(k^{2}, \chi)-c(k^{2}, \chi) = 0$. Values of ($k^{2}, \chi$) above (below) the blue dashed line correspond to $a(k^{2})b(k^{2}, \chi)-c(k^{2}, \chi)$ positive (negative). This is an example where we use the parameter set: $(a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, d) = (0.70, 0.70, 0.10, 0.30, 1.39, 0.30, 0.70, 0.25)$ Hence, $a_{1}b_{2}-rb_{1} = 0.07$, so $k_{min}^{2} = 0.26$

Figure 2.  A plot of $\chi_{c_{1}}$ and $\chi_{c_{2}}$ versus $k^{2}$. $\chi_{c_{1}}$ (red line) and $\chi_{c_{2}}$ (blue dashed line) as functions of $k^{2}$ are plotted for the parameter set: $(a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu) = (0.10, 0.20, 0.10, 1.50, 3.0, 0.6, 0.1)$

Figure 3.  Case 1: $u$ "wins": In this example the parameters are chosen so that in the pure competition system $u$ "wins". Note that while parameters are such that in the absence of a repulsive prey defense mechanism by $u$, a predator mediated coexistence is expected; yet, once this state is disrupted $v$ is driven to extinction in the presence of this taxis mechanism. Parameter set for this figure is: $(a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (0.25, 0.75, 0.75, 0.25, 0.6875, 0.5, 0.25, 15, 0.5)$

Figure 4.  Case 2: $v$ "wins": In this example the parameters are chosen so that in the pure competition system $v$ "wins". Note that while parameters are such that in the absence of a repulsive prey defense mechanism by $u$, a predator mediated coexistence is expected; yet, once this state is disrupted, the prey coexist with peak densities out of phase with the predator in the presence of this taxis mechanism. Parameter set for this figure is: $(a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (1.1, 0.1, 1.4, 0.3, 0.4, 0.4, 0.2, 50, 0.25)$

Figure 5.  Spatio-temporal patterns: $v$ "wins" case. The first column shows the initial dynamics from the perturbation of the homogeneous steady state and the second column shows the spatio-temporal pattern that is achieved. In the last row $u(x, t)$, $v(x, t)$, and $w(x, t)$ are given by the blue, green, and red lines respectively. The parameter set for this figure is: $(a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (1.25, 1.0, 0.25, 0.5, 1.25, 0.5, 0.25, 25, 0.1)$

Figure 6.  Spatio-temporal patterns: $u$ "wins" case. The first column shows the initial dynamics from the perturbation of the homogeneous steady state and the second column shows the spatio-temporal pattern that is achieved. In the last row $u(x, t)$, $v(x, t)$, and $w(x, t)$ are given by the blue, green, and red lines respectively. The parameter set for this figure is: $(a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (0.9, 1.1, 0.2, 0.2, 1.09, 0.5, 1.0, 16, 0.25)$

Figure 7.  Attracting ($\chi < 0$) case: In this example the parameters are chosen so that in the pure competition system $v$ "wins". Note that while parameters are such that in the absence of a attracing prey-taxis mechanism by $u$, a predator mediated coexistence is expected; yet, once this state is disrupted, the prey coexist with peak densities out of phase with the predator in the presence of this taxis mechanism. Parameter set for this figure is: $(a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (1.25, 1.00, 0.25, 0.50, 1.25, 0.5, 0.25, -0.1, 10)$

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