A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: I global existence and stability

IF 1.2 3区 数学 Q1 MATHEMATICS
Wonlyul Ko, Kimun Ryu
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引用次数: 0

Abstract

In this paper, we present and investigate a generalized reaction-diffusion system of predator-prey dynamics that incorporates prey-taxis and a hunting cooperation effect in predators, subject to homogeneous Neumann boundary conditions. This system describes a predator-prey interaction, in which the prey exhibit group defense mechanisms against their predators, and the predators cooperate to hunt these defended prey. The mechanism of prey is implemented through the (repulsive) prey-taxis term, which affects the diffusion rate of the predators, while the hunting cooperation effect of the predators towards their prey is implemented through the functional response. Moreover, this system incorporates generalized functional forms for the prey's growth rate, the predators' functional response and mortality rate, and the prey-tactic sensitivity, allowing for adaptation to various scenarios. We first establish that solutions of the time- and space-dependent system with such ecological characteristics exist globally and are bounded by estimating an associated weighted integral. Secondly, we investigate the constant coexistence state of the generalized system by introducing a constructed function that incorporates the prey's growth rate, the predators' functional response and mortality rate. Finally, we find some conditions yielding the local stability of all feasible constant and nonnegative solutions of the system, thereby revealing the occurrence of bistability. Furthermore, we conduct an investigation into the global stability at both the constant coexistence and predator-free states by applying Lyapunov stability analysis. We also analyze the rate at which the solutions to the system converge to these steady-states by utilizing the boundedness of the solutions along with Gagliardo-Nirenberg inequality.
捕食者与猎物--捕食者系统中捕食者与猎物--捕食者的狩猎合作:I 全局存在与稳定性
在本文中,我们提出并研究了一个捕食者-被捕食者动力学的广义反应-扩散系统,该系统在同质诺伊曼边界条件下,包含了捕食者的猎物-税收和捕食者的狩猎合作效应。该系统描述了捕食者与被捕食者之间的相互作用,在这种相互作用中,被捕食者对捕食者表现出群体防御机制,而捕食者则合作捕杀这些被防御的被捕食者。猎物机制通过影响捕食者扩散率的(排斥性)猎物-他项来实现,而捕食者对猎物的狩猎合作效应则通过功能响应来实现。此外,该系统还包含了猎物增长率、捕食者功能响应和死亡率以及猎物-战术敏感性的通用功能形式,从而可以适应各种情况。我们首先确定,具有这种生态学特征的时间和空间依赖系统的解是全局存在的,并且通过估计相关的加权积分是有界的。其次,我们通过引入一个包含猎物增长率、捕食者功能响应和死亡率的构造函数,研究了广义系统的恒定共存状态。最后,我们发现了一些条件,这些条件使得系统中所有可行的常解和非负解都具有局部稳定性,从而揭示了双稳态的发生。此外,我们还运用李亚普诺夫稳定性分析法,对恒定共存状态和无捕食者状态下的全局稳定性进行了研究。我们还利用解的有界性和 Gagliardo-Nirenberg 不等式分析了系统解收敛到这些稳定状态的速度。
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来源期刊
CiteScore
2.50
自引率
7.70%
发文量
790
审稿时长
6 months
期刊介绍: The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Papers are sought which employ one or more of the following areas of classical analysis: • Analytic number theory • Functional analysis and operator theory • Real and harmonic analysis • Complex analysis • Numerical analysis • Applied mathematics • Partial differential equations • Dynamical systems • Control and Optimization • Probability • Mathematical biology • Combinatorics • Mathematical physics.
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