Invasion Traveling Waves for a Discrete Diffusive Ratio-Dependent Predator-Prey Model
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
INVASION TRAVELING WAVES FOR A DISCRETE DIFFUSIVE RATIO-DEPENDENT PREDATOR-PREY MODEL∗
7)
Tao SU (
ÜI)
Guobao ZHANG (
†
Colloge of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China E-mail : [email protected]; [email protected] Abstract This article is concerned with the existence of traveling wave solutions for a discrete diffusive ratio-dependent predator-prey model. By applying Schauder’s fixed point theorem with the help of suitable upper and lower solutions, we prove that there exists a positive constant c∗ such that when c > c∗ , the discrete diffusive predator-prey system admits an invasion traveling wave. The existence of an invasion traveling wave with c = c∗ is also established by a limiting argument and a delicate analysis of the boundary conditions. Finally, by the asymptotic spreading theory and the comparison principle, the non-existence of invasion traveling waves with speed c < c∗ is also proved. Key words
predator-prey system; ratio-dependent functional response; discrete diffusion; invasion traveling waves
2010 MR Subject Classification
1
34A33; 34K60; 92D25
Introduction
In this article, we consider a discrete diffusive Holling-Tanner type predator-prey system with ratio-dependent functional response duj (t) Auj (t)vj (t) dt = D[uj ](t) + uj (t)(1 − uj (t)) − u (t) + av (t) , j j (1.1) dv (t) v (t) j j = dD[vj ](t) + Bvj (t) 1 − , dt uj (t) where j ∈ Z, t > 0, d > 0 and
D[zj ] = zj+1 + zj−1 − 2zj . Here, uj (t) and vj (t) are the population densities of the prey and predator species at niche j and time t, respectively. A, a, d and B are positive constants. The parameter d > 0 is a rescaled diffusion coefficient of the predator species while the diffusion coefficient for the prey Auj (t) is rescaled to be 1. The functional response of predator to prey is given by uj (t)+av , which j (t) ∗ Received
October 29, 2018; revised February 24, 2020. This work was supported by NSF of China (11861056), Gansu Provincial Natural Science Foundation (18JR3RA093). † Corresponding author: Guobao ZHANG.
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
is ratio-dependent Holling type II. The parameter A is the capturing rate, and a is the halfcapturing saturation constant. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of the predator species follows a logistic dynamic with a varying carrying capacity proportional to the density of prey. B denotes the intrinsic growth rate of the predator. It is easy to see that system (1.1) has two nonnegative spatially constant equilibria (1, 0) and (k, k), where k = 1 − A/(a + 1) when A < a + 1. Clearly, (1, 0) is unstable and (k, k) is stable. This implies that we can observe the invading coexistence phenomenon between the resident (the prey) and the invader (the predator). To describe such an invading coexistence phenomenon, the traveling wave solution play
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