Existence and Uniqueness of the Positive Steady State Solution for a Lotka-Volterra Predator-Prey Model with a Crowding
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
EXISTENCE AND UNIQUENESS OF THE POSITIVE STEADY STATE SOLUTION FOR A LOTKAVOLTERRA PREDATOR-PREY MODEL WITH A CROWDING TERM∗
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Xianzhong ZENG (
†
4 Z)
Lingyu LIU (
)
Weiyuan XIE (
School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan 411201, China E-mail : [email protected]; [email protected]; [email protected] Abstract This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation. We obtain a critical value λD 1 (Ω0 ), and demonstrate that the existence of the predator in Ω0 only depends on the relationship of the growth rate µ of D the predator and λD 1 (Ω0 ), not on the prey. Furthermore, when µ < λ1 (Ω0 ), we obtain the existence and uniqueness of its positive steady state solution, while when µ ≥ λD 1 (Ω0 ), the predator and the prey cannot coexist in Ω0 . Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding region Ω0 , which is different from that of the classical Lotka-Volterra predator-prey model. Key words
Lotka-Volterra predator-prey model; crowding term; critical value; coexistence
2010 MR Subject Classification
1
35J57; 35B25; 92C40
Introduction
In the spatial population models, the heterogeneous factors of the environment are often not negligible. In this case, some parameters of the corresponding population models, such as the growth rates, the environment capacities and the coefficients of population interactions, will become the functions of the space variable x. Thus, the predator-prey models in a spatially heterogeneous environment usually take the form u ut − d1 △u = λ(x)u(1 − k (x) ) − b(x)f (u, v)v, x ∈ Ω, t > 0; 1 (1.1) v ) + c(x)f (u, v)v, x ∈ Ω, t > 0. vt − d2 △v = µ(x)v(1 − k2 (x) To our knowledge, many pioneers showed that the behaviors of the solutions of the population models in a spatially heterogeneous environment are very sensitive to the change of ∗ Received April 27, 2019; revised September 15, 2019. The work was supported by the Hunan Provincial Natural Science Foundation of China (2019JJ40079, 2019JJ50160), the Scientific Research Fund of Hunan Provincial Education Department (16A071, 19A179) and the National Natural Science Foundation of China (11701169) † Corresponding author: Xianzhong ZENG.
1962
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
certain coefficient functions in part of the underlying spatial region. For example, for the logistic model ∂t u − △u = λu − a(x)u2 in Ω, where Ω0 is an open and connected subset of Ω such that a(x) = 0 in Ω0 and a(x) > 0 in Ω \ Ω0 , Cirstea and Radulescu[3–5], Du and Huang[16, 17], J. M. Fraile and his coauthors [18], Lopez-Gomez and his coauthors [22, 24–26] and Ouyang [31] etc obtained a critical patch size described by the principal eigenvalue λD 1 (Ω0 ), and demonD strated that if λ ≥ λD (Ω ), then lim u(x, t) = ∞ in Ω , while if 0 < λ < λ 0 0 1 1 (Ω0 ), then u(x, t) t
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