A delayed lattice dynamical system with non-local diffusion and interaction is considered in this paper. The exact asymptotics of the wave profile at both wave tails is derived, and all the wave profiles are shown to be strictly increasing. Moreover, we prove that the wave profile with a given admissible speed is unique up to translation. These results generalize earlier monotonicity, asymptotics and uniqueness results in the literature.
The article is dedicated to the dynamics of a stochastic SIRS epidemic model, which is obtained by introducing Gaussian white noise to the transmission coefficient of a deterministic epidemic model with non-monotone and saturated incidence rate. The existence and uniqueness of positive global solution is proved for the stochastic model. The threshold parameter R is established, and under some acceptable conditions the disease will go to extinction if R < 1. However, the stochastic system has a unique ergodic stationary distribution and the disease is persistent if R > 1. We also analyze the asymptotic behavior of the stochastic model near the disease-free equilibrium of the corresponding deterministic system. Numerical simulation is provided to support our theoretical results.
In this paper, we investigate the dynamics of a stochastic predator-prey model with ratio-dependent functional response and disease in the prey. Firstly, we prove the existence and uniqueness of the positive solution for the stochastic model by using conventional methods. Then we obtain the threshold 0 s R for the infected prey population, that is, the disease will tend to extinction if 0 1 s R < , and it will exist in the long time if 0 1 s R > . Finally, the sufficient condition on the existence of a unique ergodic stationary distribution is obtained, which indicates that all the populations are permanent in the time mean sense. Numerical simulations are conducted to verify our analysis results.
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