# American Institute of Mathematical Sciences

October  2017, 10(5): 1025-1042. doi: 10.3934/dcdss.2017054

## Pattern dynamics of a delayed eco-epidemiological model with disease in the predator

 1 Department of Computer Science and Technology, North University of China, Taiyuan Shan'xi 030051, China 2 Complex Systems Research Center, Shanxi University, Taiyuan Shan'xi 030051, China

* Corresponding author: Zhen Jin

Received  October 2016 Revised  January 2017 Published  June 2017

Fund Project: The work is supported by the National Natural Science Foundation of China under Grants (11331009,11671241 and 11301490), 131 Talents of Shanxi University, Program for the Outstanding Innovative Teams (OIT) of Higher Learning Institutions of Shanxi, and Natural Science Foundation of Shanxi Province Grant no. 201601D021002

The eco-epidemiology, combining interacting species with epidemiology, can describe some complex phenomena in real ecosystem. Most diseases contain the latent stage in the process of disease transmission. In this paper, a spatial eco-epidemiological model with delay and disease in the predator is studied. By mathematical analysis, the characteristic equations are derived, then we give the conditions of diffusion-driven equilibrium instability and delay-driven equilibrium instability, and find the ranges of existence of Turing patterns in parameter space. Moreover, numerical results indicate that a parameter variation has influences on time and spatially averaged densities of pattern reaching stationary states when other parameters are fixed. The obtained results may explain some mechanisms of phenomena existing in real ecosystem.

Citation: Jing Li, Zhen Jin, Gui-Quan Sun, Li-Peng Song. Pattern dynamics of a delayed eco-epidemiological model with disease in the predator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1025-1042. doi: 10.3934/dcdss.2017054
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##### References:
Schematic diagrams of the cubic function $y(e)$ for $y_{1}>0$ in Theorem 3.3. (a) $y_{3}x y0$. (b) $y_{3}=0$ and $y_{2} < 0$. (c) $y_{3}>0$, $y_{2}^{2}-3y_{1}y_{3}>0$ and $y_{2} < 0$
The bifurcation diagram of system (4) in parameter space $r-h$. (a) Parameters are $\beta_{1}=1.8$, $\mu=0.6$, $m=0.8$, $D_{1}=1$, $D_{2}=0.03$, $D_{3}=2$, $\tau=0.01$. (b) Parameters are $\beta_{1}=1.8$, $\mu=0.6$, $m=0.7$, $D_{1}=10$, $D_{2}=0.1$, $D_{3}=4$, $\tau=0.01$
Coefficients of the dispersion relation of the characteristic equation (16) for $r=0.1$, $h=0.07$, $\beta_{1}=1.8$, $\mu=0.6$, $m=0.8$, $D_{1}=1$, $D_{2}=0.03$, $D_{3}=2$, $\tau=0.01$
Coefficients of the dispersion relation of the characteristic equation (16) for $r=0.1$, $h=0.1$, $\beta_{1}=1.8$, $\mu=0.6$, $m=0.7$, $D_{1}=10$, $D_{2}=0.1$, $D_{3}=4$, $\tau=0.01$
Schematic diagrams of the cubic function $y(e)$ for $y_{1}>0$ in Theorem 3.4. (a) $y_{2}^{2}-3y_{1}y_{3}\leq 0$. (b) $y_{3}>0$, $y_{2}^{2}-3y_{1}y_{3}>0$ and $y_{2}>0$. (c) $y_{3}=0$ and $y_{2}>0$. (d) $y_{3} < 0$. (e) $y_{3}=0$ and $y_{2} < 0$. (f) $y_{3}>0$, $y_{2}^{2}-3y_{1}y_{3}>0$ and $y_{2} < 0$
Spatial patterns (top) and the corresponding spatially averaged population density (bottom). (a) Small "black-eye" pattern (r=0.1), (b) small "black-eye" pattern (r=0.15)
Spatial patterns (top) and the corresponding spatially averaged population density (bottom). (a) Big "black-eye" pattern (h=0.1), (b) big "black-eye" pattern (h=0.14)
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