Bifurcations of a delayed predator-prey system with fear, refuge for prey and additional food for predator.

Taking into account the impacts of the fear by predator, anti-predation response, refuge for prey, additional food supplement for predator and the delayed fear induced by the predator, we establish a delayed predator-prey model in this paper. We analyze the persistence and extinction of species and the existence and uniqueness of a coexistence fixed point. Particularly, we investigate the local asymptotic stability of the equilibrium by use of the characteristic equation theory of a variational matrix. Applying the Hopf bifurcation theorem, we investigate and obtain the bifurcation thresholds of the parameters of fear, refuge coefficient, the quality and quantity of additional food and the anti-predation delayed response produced by prey. Finally we give some examples to verify our theoretical findings and clarify the detailed influences of these parameters on the system dynamics. The main conclusions reveal that these parameters play an important role in the long-term behaviors of species and should be applied correctly to preserve the continuous development of species.

Bifurcations of a prey-predator system with fear, refuge and additional food.

In the predator-prey system, predators can affect the prey population by direct killing and inducing predation fear, which ultimately force preys to adopt some anti-predator strategies. Therefore, it proposes a predator-prey model with anti-predation sensitivity induced by fear and Holling-Ⅱ functional response in the present paper. Through investigating the system dynamics of the model, we are interested in finding how the refuge and additional food supplement impact the system stability. With the changes of the anti-predation sensitivity (the refuge and additional food), the main result shows that the stability of the system will change accordingly, and it has accompanied with periodic fluctuations. Intuitively the bubble, bistability phenomena and bifurcations are found through numerical simulations. The bifurcation thresholds of crucial parameters are also established by the Matcont software. Finally, we analyze the positive and negative impacts of these control strategies on the system stability and give some suggestions to the maintaining of ecological balance, we perform extensive numerical simulations to illustrate our analytical findings.

On the System of High Order Rational Difference Equations.

This paper is concerned with the boundedness, persistence, and global asymptotic behavior of positive solution for a system of two rational difference equations x n+1 = A + (x n /∑ i=1 (k) y n-i ), y n+1 = B + (y n /∑ i=1 (k) x n-i ), n = 0,1,…, k ∈ {1,2,…}, where A, B ∈ (0, ∞), x -i ∈ (0, ∞), and y -i ∈ (0, ∞), i = 0,1, 2,…, k.