2015/05/25 by Xiao He, Sining Zheng, He, Xiao +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #35J47 #35K57 #92D40 #Analysis of PDEs (math.AP) #Evolution and Genetic Dynamics #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models
paper · pdf · doi:10.48550/arxiv.1505.06625
openalex publication_date 2015/05/25 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
In any reaction-diffusion system of predator-prey models, the population densities of species are determined by the interactions between them, together with the influences from the spatial environments surrounding them. Generally, the prey species would die out when their birth rate is too low, the habitat size is too small, the predator grows too fast, or the predation pressure is too high. To save the endangered prey species, some human interference is useful, such as creating a protection zone where the prey could cross the boundary freely but the predator is prohibited from entering. This paper studies the existence of positive steady states to a predator-prey model with reaction-diffusion terms, Beddington-DeAngelis type functional response and non-flux boundary conditions. It is shown that there is a threshold value θ0 which characterizes the refuge ability of prey such that the positivity of prey population can be ensured if either the prey's birth rate satisfies θ≥θ0 (no matter how large the predator's growth rate is) or the predator's growth rate satisfies μ≤ 0, while a protection zone Ω0 is necessary for such positive solutions if θ0 properly large. The more interesting finding is that there is another threshold value θ^*=θ^*(μ,Ω0)θ1(Ω0), prey species could survive no matter how large the predator's growth rate is. In addition, we get the fourth threshold value θ_* for negative μ such that the system admits positive steady states if and only if θ>θ_*.