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International Journal of Biomathematics 2024年第1期17卷 81-112页

作者： Liliana Puchuri Orestes Bueno Pontificia Universidad Catolica del Peri Av.Universitaria 1801 San MiguelLima 15088Peru Universidad del Pacifico Jr.Gral Luis Sanchez Cerro 2141 Jesus MariaLima 15072Peru

In this work,we study a predator-prey model of Gause type,in which the prey growth rate is subject to an allee effect and the action of the predator over the prey is determined by a generalized hyperbolic-type functional response,which is neither differentiable nor locally Lipschitz at the predator axis.This kind of functional response is an extension of the so-called square root functional response,used to model systems in which the prey have a strong herd structure.We study the behavior of the solutions in the first quadrant and the existence of limit cycles.We prove that,for a wide choice of parameters,the solutions arrive at the predator axis in finite time.We also characterize the existence of an equilibrium point and,when it exists,we provide necessary and sufficient conditions for it to be a center-type equilibrium.In fact,we show that the set of parameters that yield a center-type equilibrium,is the graph of a function with an open domain.We also prove that any center-type equilibrium is stable and it always possesses a supercritical Hopf bifurcation.In particular,we guarantee the existence of a unique limit cycle,for small perturbations of the system.

关键词： Predator-prey models Gause models allee effect non-differentiable functional response

Dynamical behaviors of a constant prey refuge ratio-dependent prey-predator model with allee and fear effects

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International Journal of Biomathematics 2024年第1期17卷 227-250页

作者： Soumitra Pal Pijush Panday Nikhil Pal A.K.Misra Joydev Chattopadhyays Department of Mathematics Institute of Science Banaras Hindu University Varanasi 221005India Department of Mathematical and Statistical Sciences University of Alberta EdmontonAB T6G 2G1Canada Department of Mathematics Visva-Bharati Santiniketan 731235India Agricultural and Ecological Research Unit Indian Statistical Institute 203 B.T.RoadKolkata 700108India

In this paper,we consider a nonlinear ratio-dependent prey-predator model with constant prey refuge in the prey population.Both allee and fear phenomena are incorporated explicitly in the growth rate of the prey population.The qualitative behaviors of the proposed model are investigated around the equilibrium points in detail.Hopf bifurcation including its direction and stability for the model is also studied.We observe that fear of predation risk can have both stabilizing and destabilizing effects and induces bubbling phenomenon in the system.It is also observed that for a fixed strength of fear,an increase in the allee parameter makes the system unstable,whereas an increase in prey refuge drives the system toward stability.However,higher values of both the allee and prey refuge parameters have negative impacts and the populations go to extinction.Further,we explore the variation of densities of the populations in different bi-parameter spaces,where the coexistence equilibrium point remains stable.Numerical simulations are carried out to explore the dynamical behaviors of the system with the help of MATLAB software.

关键词： Predator-prey system allee effect fear effect prey refuge bifurcation population density

effect of random movement and cooperative hunting in the prey-predator system:A dynamical approach

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International Journal of Biomathematics 2024年第3期17卷 211-240页

作者： Shivam Teekam Singh Mukesh Kumar Department of Mathematics Graphic Era Deemed to be University 566/6 Bell RoadSociety AreaClenent Town DehradunUttarakhand248002India Department of Computer Science and Engineering Graphic Era Deemed to be University 566/6 Bell RoadSociety AreaClement Town DehradunUttarakhand248002India

Self-diffusion prerequisite is obtained as the spreading approach of biological populations.Cooperative hunting is a common behavior in predator populations that promotes predation and the coexistence of the prey-predator system.On the other side,the allee effect among prey may cause the system to become unstable.In this paper,a difusive prey predator system with cooperative hunting and the weak allee effect in prey populations is discussed.The linear stability and Hopf-bifurcation analysis had been used to examine the system's stability.From the spatial stability of the system,the conditions for Turing instability have been derived.The multiple-scale analysis has been used to derive the amplitude equations of the system.The stability analysis of these amplitude equations leads to the formation of Turing patterns.Finally,numerical simulations are used to analyze spatial patterns forming in 1-D and 2-D.The studies indicate that the model can generate a complex pattern structure and that self-diffusion has a drastic impacton species distribution.

关键词： Prey predator system hunting cooperation allee effect Hopf-bifurcation diffusive instability amplitudeequation

BIFURCATION ANALYSIS IN A PREDATOR-PREY MODEL WITH AN allee effect AND A DELAYED MECHANISM

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Acta Mathematica Scientia 2023年第3期43卷 1415-1438页

作者：李丹阳刘华张濠天马明叶勇魏玉梅 School of Mathematics and Computer Science Northwest Minzu UniversityLanzhou730000China School of Science Harbin Institute of Technology(Shenzhen)Shenzhen518055China Experimental Teaching Department Northwest Minzu UniversityLanzhou730030China

Regarding delay-induced predator-prey models, much research has been done on delayed destabilization, but whether delays are stabilizing or destabilizing is a subtle issue. In this study, we investigate predator-prey dynamics affected by both delays and the allee effect. We analyze the consequences of delays in different feedback mechanisms. The existence of a Hopf bifurcation is studied, and we calculate the value of the delay that leads to the Hopf bifurcation. Furthermore, applying the normal form theory and a center manifold theorem, we consider the direction and stability of the Hopf bifurcation. Finally, we present numerical experiments that validate our theoretical analysis. Interestingly, depending on the chosen delay mechanism, we find that delays are not necessarily destabilizing. The allee effect generally increases the stability of the equilibrium, and when the allee effect involves a delay term, the stabilization effect is more pronounced.

关键词： delays allee effect Hopf bifurcation stability

Stability and Turing Patterns of a Predator-prey Model with Holling Type Ⅱ Functional Response and allee effect in Predator

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Acta Mathematicae Applicatae Sinica 2023年第3期39卷 675-695页

作者： Lu CHEN Feng YANG Yong-li SONG School of Mathematics Hangzhou Normal UniversityHangzhou 311121China

In this paper, we are concerned with a predator-prey model with Holling type Ⅱ functional response and allee effect in predator. We first mathematically explore how the allee effect affects the existence and stability of the positive equilibrium for the system without diffusion. The explicit dependent condition of the existence of the positive equilibrium on the strength of allee effect is determined. It has been shown that there exist two positive equilibria for some modulate strength of allee effect. The influence of the strength of the allee effect on the stability of the coexistence equilibrium corresponding to high predator biomass is completely investigated and the analytically critical values of Hopf bifurcations are theoretically determined.We have shown that there exists stability switches induced by allee effect. Finally, the diffusion-driven Turing instability, which can not occur for the original system without allee effect in predator, is explored, and it has been shown that there exists diffusion-driven Turing instability for the case when predator spread slower than prey because of the existence of allee effect in predator.

关键词： predator-prey model allee effect diffusion stability Turing bifurcation

Hopf bifurcation of a delayed predator-prey model with allee effect and anti-predator behavior

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生物数学学报:英文版 2023年第7期16卷 111-139页

作者： Xinyue Xu Yan Meng Yangyang Shao School of Mathematics and Physics University of Science and Technology Beijing Beijing 100083P.R.China

This paper proposes a diffusive predator-prey model with allee effect,time delay and anti-predator behavior.First,the existence and stability of all equilibria are analyzed and the conditions for the appearance of the Hopf bifurcation are studied.Using the normal form and center manifold theory,the formulas which can determine the direction,period and stability of Hopf bifurcation are obtained.Numerical simulations show that the allee effect can determine the survival abundance of the prey and predator populations,and anti-predator behavior can greatly improve the stability of the coexisting equilibrium.

关键词： Predator-prey Hopf bifurcation delay and diffusion allee effect numerical simulation

Existence and nonexistence of positive solutions of semilinear elliptic equation with inhomogeneous strong allee effect

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Applied Mathematics and Mechanics(English Edition) 2009年第11期30卷 1461-1468页

作者：刘冠琦王玉文史峻平 Y. Y. Tseng Functional Analysis Research Center and Department of Mathematics Harbin Normal University Department of Mathematics College of William and Mary

In this paper, we study a semilinear elliptic equation defined on a bounded smooth domain. This type of problem arises from the study of spatial ecology model, and the growth function in the equation has a strong allee effect and is inhomogeneous. We use variational methods to prove that the equation has at least two positive solutions for a large parameter if it satisfies some appropriate conditions. We also prove some nonexistence results.

关键词： semilinear equation allee effect positive solutions existence

Dynamic Analysis of an Algae-Fish Harvested Model with allee effect

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应用数学与应用物理(英文) 2023年第10期11卷 2938-2962页

作者： Xiaoyu Song Hengguo Yu Min Zhao School of Mathematics and Physics Wenzhou University Wenzhou China Key Laboratory for Subtropical Oceans & Lakes Environment and Biological Resources Utilization Technology of Zhejiang Wenzhou University Wenzhou China School of Life and Environmental Science Wenzhou University Wenzhou China

In this paper, an algae-fish harvested model with allee effect was established to further explore the dynamic evolution mechanism under the influence of key factors. Mathematical theoretical work not only investigated the existence and stability of all possible equilibrium points, but also probed into the occurrence of transcritical and Hopf bifurcation. The numerical simulation works verified the effectiveness of the theoretical derivation results and displayed rich bifurcation dynamical behaviors, which showed that allee effect and harvest have played a vital role in the dynamic relationship between algae and fish. In summary, it was expected that these research results would be beneficial for promoting the study of bifurcation dynamics in aquatic ecosystems.

关键词： Algae and Fish Model allee effect Harvesting Equilibrium Point Bifurcation

Dynamics Analysis of an Aquatic Ecological Model with allee effect

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Applied Mathematics 2022年第10期13卷 822-844页

作者： Lingzhen Zheng Jueyuan Yan Hengguo Yu School of Mathematics and Physics Wenzhou University Wenzhou China

In this paper, based on the dynamic relationship between algae and protozoa, an aquatic ecological model with allee effect was established to investigate how some ecological environment factors affect coexistence mode of algae and protozoa. Mathematical derivation works mainly gave some key conditions to ensure the existence and stability of all possible equilibrium points, and to induce the occurrence of transcritical bifurcation and Hopf bifurcation. The numerical simulation works mainly revealed ecological relationship change characteristics of algae and protozoa with the help of bifurcation dynamics evolution process. Furthermore, it was also worth emphasizing that allee effect had a strong influence on the dynamic relationship between algae and protozoa. In a word, it was hoped that the research results could provide some theoretical support for algal bloom control, and also be conducive to the rapid development of aquatic ecological models.

关键词： Algae Protozoa allee effect Bifurcation Relationship

Dynamics in a Discrete-time Predator-prey System with allee effect

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Acta Mathematicae Applicatae Sinica 2013年第1期29卷 143-164页

作者： Xian-wei Chen Xiang-ling Fu Zhu-jun Jing Department of Mathematics Hunan Normal University School of Mathematics and Computational Science Hunan University of Science and Technology Academy of Mathematics and Systems Science Chinese Academy of Sciences

In this paper, dynamics of the discrete-time predator-prey system with allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory, and then further illustrated by numerical simulations. Chaos in the sense of Marotto is proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and rich dynamical behavior. More specifically, apart from stable dynamics, this paper presents the finding of chaos in the sense of Marotto together with a host of interesting phenomena connected to it. The analytic results and numerical simulations demostrates that the allee constant plays a very important role for dynamical behavior. The dynamical behavior can move from complex instable states to stable states as the allee constant increases （within a limited value）. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the discrete-time predator-prey with allee effect is given.

关键词： Predator-prey System allee effect flip bifurcation Hopf bifurcation Marotto＇s chaos transientchaos invariant circle periodic window