Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis

Ke Wang; Qi Wang; Feng Yu

doi:10.3934/dcds.2017021

Article Contents

2017, Volume 37, Issue 1: 505-543. Doi: 10.3934/dcds.2017021

This issue Previous Article Monotone dynamical systems: Reflections on new advances & applications Next Article Random attractor for stochastic non-autonomous damped wave equation with critical exponent

Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis

Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China

*Corresponding author
*Corresponding author

^†currently at Department of Mathematics, University of Central Florida, USA

Received: December 2015

Revised: August 2016

Published online: January 2017

QW is supported by NSF-China (Grant No. 11501460) and the Project (No.15ZA0382) from Department of Education, Sichuan, China.

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Abstract

This paper concerns pattern formation in a class of reaction-advection-diffusion systems modeling the population dynamics of two predators and one prey. We consider the biological situation that both predators forage along the population density gradient of the preys which can defend themselves as a group. We prove the global existence and uniform boundedness of positive classical solutions for the fully parabolic system over a bounded domain with space dimension $ N=1,2 $ and for the parabolic-parabolic-elliptic system over higher space dimensions. Linearized stability analysis shows that prey-taxis stabilizes the positive constant equilibrium if there is no group defense while it destabilizes the equilibrium otherwise. Then we obtain stationary and time-periodic nontrivial solutions of the system that bifurcate from the positive constant equilibrium. Moreover, the stability of these solutions is also analyzed in detail which provides a wave mode selection mechanism of nontrivial patterns for this strongly coupled system. Finally, we perform numerical simulations to illustrate and support our theoretical results.

Keywords:

Mathematics Subject Classification: Primary:35B36, 92D25, 35B10, 35B32;Secondary:35B35, 35J47, 35K20, 35Q92.

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Figure 1. Pitch-fork bifurcation diagrams when case (ⅰ) in Theorem 4.3 occurs. The stable bifurcation curve is plotted in solid line and the unstable bifurcation curve is plotted in dashed line. The branch $\Gamma_{k_0}(s)$ around $(\bar u, \bar v, \bar w, \chi^S_{k_0})$ is stable if it turns to the right and is unstable if it turns to the left, while $\Gamma_{k}(s)$ around $(\bar u, \bar v, \bar w, \chi^S_{k})$ is always unstable if $k\neq k_0$

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Figure 2. Formation of stationary patterns of (3.1) over $\Omega=(0, 7)$. System parameters in all the graphes are taken to be the same as in Table 1 except that $\chi=8$, which is larger than $\chi^S_{k_0}\approx6.05$ given in Table 1. Initial data are $(u_0, v_0, w_0)=(\bar u, \bar v, \bar w)+(0.01, 0.01, 0.01)\cos \pi x$, while the stable pattern has wave mode $\cos \frac{6\pi x}{7}$. These graphes support our stability analysis of the bifurcating solutions

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Figure 3. Formation of stationary patterns of (3.1) over intervals with lengthes $L=9$, $11$, $13$ and $15$. System parameters here are taken to be the same as those in Table 1 except that $\chi=8$, which is slightly larger than $\chi^S_{k_0}\approx6.04$ given in Table 2. Initial data are $(u_0, v_0, w_0)=(\bar u, \bar v, \bar w)+(0.01, 0.01, 0.01)\cos \pi x$. These graphes support our stability analysis of the bifurcating solutions and indicate that large intervals support more aggregates than small intervals

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Figure 4. Formation of time-periodic spatial patterns of (3.1) over $\Omega=(0, 7)$. System parameters in all the graphes are taken to be the same as in Table 3 except that $\chi=120$, which is slightly larger than $\chi^H_{k_0}\approx 92.57$ given in Table 3. Initial data are $(u_0, v_0, w_0)=(\bar u, \bar v, \bar w)+(0.01, 0.01, 0.01)\cos \pi x$, however the stable oscillating patterns have spatial profile $\cos\frac{3\pi x}{7}$, which emerge periodically. These plots support our stability analysis in Section 5

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Figure 5. Formation of time-periodic spatial patterns of (3.1) over intervals with lengthes $L=9$, $11$, $13$ and $15$ respectively. System parameters in all the graphes are taken to be the same as in Table 3 except that $\chi=120$, which is slightly larger than $\chi^H_{k_0}$ given in Table 4. Initial data are $(u_0, v_0, w_0)=(\bar u, \bar v, \bar w)+(0.01, 0.01, 0.01)\cos \pi x$. These graphes support our stability analysis of the bifurcating solutions

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Figure 6. Formation and development of boundary spike through wave propagation. Initial data are $(u_0, v_0, w_0)=(\bar u, \bar v, \bar w)+(0.01, 0.01, 0.01)\cos \pi x$. Prey-taxis rate $\chi=3000$ which is far away from the critical bifurcation value $\chi_0=956.79$.

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Figure 7. Pattern formations in (3.1) due to the effect of large prey-taxis rate $\chi$. Various interesting and complex spatial-temporal dynamics are observed in this system

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Table 1. Values of $\chi^S_k$ in (3.4) and $\chi^H_k$ in (3.5) for $L=7$. The system parameters are chosen to be $d_1=d_3=0.1$, $d_2=2$, $\alpha_1=\beta_1=\beta_2=0.5$, $\alpha_2=2$, $\alpha_3=1$ and $\beta_{31}=\beta_{32}=0.1$, $\xi=0.5$. The sensitivity function is $\phi(w)=w(0.1-w)$ which models the group defense of the preys when population density surpasses 0.1. We see that $\min_{k\in\mathbb N^+}\{\chi^S_k, \chi^H_k\}=\chi^S_6$. Therefore $(\bar u, \bar v, \bar w)$ loses its stability to the stable wave mode $\cos \frac{6\pi x}{7}$. This is numerically verified in Figure 2

$k$	1	2	3	4	5	6	7	8	9
$\chi^S_{k}$	66.98	18.98	10.30	7.49	6.41	6.05	6.09	6.36	6.80
$\chi^H_{k}$	1204.20	550.84	504.20	575.80	705.50	878.48	1089.70	1336.90	1619.19

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Table 2. Stable wave mode numbers and the corresponding bifurcation values $\chi_0$ for different interval lengthes. System parameters are chosen to be the same as those in Table 1. We see that the threshold value $\chi_0$ is always achieved at the steady state bifurcation point $\chi^S_{k_0}$. This table also indicates that larger intervals support higher wave modes

Interval length $L$	1	2	3	4	5	6	7	8
$k_0$	1	2	3	4	5	5	6	7
$\chi_0=\chi^S_{k_0}$	6.09	6.09	6.09	6.09	6.09	6.08	6.05	6.04
Interval length $L$	9	10	11	12	13	14	15	16
$k_0$	8	9	10	11	12	13	14	15
$\chi_0=\chi^S_{k_0}$	6.04	6.03	6.04	6.04	6.04	6.04	6.04	6.04

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Table 3. Values of $\chi^S_k$ in (3.4) and $\chi^H_k$ in (3.5) for $L=7$. System parameters are $d_1=d_3=1$, $d_2=0.01$, $\alpha_1=0.02$, $\alpha_2=0.04$, $\alpha_3=8$ and $\beta_1=0.05$, $\beta_2=\beta_{31}=\beta_{32}=0.5$, while the sensitivity function is $\phi(w)=w(0.2-w)$. We see that $\min_{k\in\mathbb N^+}\{\chi^S_k, \chi^H_k\}=\chi^H_3$. Therefore $(\bar u, \bar v, \bar w)$ loses its stability to the time-periodic solutions with wave mode $\cos \frac{3\pi x}{7}$. This is numerically verified in Figure 4

$k$	1	2	3	4	5	6	7	8	9
$\chi^S_{k}$	106.4	98.63	107.32	122.53	143.15	169.05	200.24	236.80	278.76
$\chi^H_{k}$	186.37	96.73	92.57	105.46	127.03	155.24	189.40	229.24	274.64

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Table 4. Stable wave mode numbers and the corresponding bifurcation values $\chi_0$ for different interval lengthes, where system parameters are chosen to be the same as those in Table 3. We see that the threshold value $\chi_0$ is always achieved at the Hopf bifurcation point $\chi^H_{k_0}$

Interval length $L$	2	3	4	5	6	7	8	9
$k_0$	1	2	3	4	5	5	6	7
$\chi_0=\chi^H_{k_0}$	97.68	92.13	97.68	91.49	92.13	92.57	91.15	92.13
Interval length $L$	10	11	12	13	14	15	16	17
$k_0$	8	9	10	11	12	13	14	15
$\chi_0=\chi^H_{k_0}$	91.5	91.2	91.13	91.21	91.30	91.49	91.15	91.40

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