In this paper,we establish a delayed predator-prey model with nonlocal fear effect.Firstly,the existence,uniqueness,and persistence of solutions of the model are studied.Then,the local stability,Turing bifurcation,and...In this paper,we establish a delayed predator-prey model with nonlocal fear effect.Firstly,the existence,uniqueness,and persistence of solutions of the model are studied.Then,the local stability,Turing bifurcation,and Hopf bifurcation of the constant equilibrium state are analyzed by examining the characteristic equation.The global asymptotic stability of the positive equilibrium point is investigated using the Lyapunov function method.Finally,the correctness of the theoretical analysis results is verified through numerical simulations.展开更多
In this paper, the dynamical behaviors of a modified Leslie-Gower predator-prey model incorporating fear effect and prey refuge are investigated. We delve into the construction of the model and its biological signific...In this paper, the dynamical behaviors of a modified Leslie-Gower predator-prey model incorporating fear effect and prey refuge are investigated. We delve into the construction of the model and its biological significance, with preliminary results encompassing positivity, boundedness, and persistence. The stability of the system’s boundary and positive equilibrium points is proven by calculating the real part of the eigenvalues of the Jacobian matrix. At the positive equilibrium point, we demonstrate that the system’s unique positive equilibrium is globally asymptotically stable by using the Dulac criterion. Furthermore, at this equilibrium point, we employ the Implicit Function Theorem to discuss how fear effects and prey refuges influence the population densities of both prey and predators. Finally, numerical simulations are conducted to validate the above-mentioned conclusions and explored the impact of Predator-taxis sensitivity αon dynamics of the system.展开更多
In this manuscript, we have studied a fractional-order tri-trophic model with the help of Caputo operator. The total population is divided into three parts, namely prey, intermediate predator and top predator. In addi...In this manuscript, we have studied a fractional-order tri-trophic model with the help of Caputo operator. The total population is divided into three parts, namely prey, intermediate predator and top predator. In addition, the predator fear impact on prey population is suggested in this paper. Existence and uniqueness along with non-negativity and boundedness of the model system have been investigated. We have studied the local stability at all equilibrium points. Also, we have discussed global stability and Hopf bifurcation of our suggested model at interior equilibrium point. The Adam-Bashforth-Moulton approach is utilized to approximate the solution to the proposed model. With the help of MATLAB, we were able to conduct graphical demonstrations and numerical simulations.展开更多
In this research,we examine a predator–prey model in which nonlocal fear plays a role alongside delay in a reaction–diffusion framework.We integrate two delays into the model to account for the lag between when fear...In this research,we examine a predator–prey model in which nonlocal fear plays a role alongside delay in a reaction–diffusion framework.We integrate two delays into the model to account for the lag between when fear starts affecting the growth rate of prey and when it starts affecting the growth rate of the predator through feedback.The first step is to investigate local and global stability and bifurcations in the equilibrium states of the nondelayed model.We explore the Hopf bifurcation in the delayed model using the delay as the bifurcation parameter.Our theoretical findings are then backed up by certain simulations.It reveals how the system,depending on its level of anxiety and the time delays involved,displays a wide range of spatiotemporal patterns.展开更多
In this paper, we consider the fear effect and gestation delay, and then establish a delayed predator-prey model with cannibalism. Firstly, we prove the well-posedness of the model. Secondly, the existence and stabili...In this paper, we consider the fear effect and gestation delay, and then establish a delayed predator-prey model with cannibalism. Firstly, we prove the well-posedness of the model. Secondly, the existence and stability of all equilibriums of the system are studied. Thirdly, the Hopf bifurcation at the coexistence equilibrium is investigated, and the conditions for the occurrence of Hopf bifurcation at the unique positive equilibrium point of the system with delay are determined. Finally, the numerical simulation results show that as the time delay increases, the equilibrium loses its stability, and the system has periodic solution.展开更多
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 popul...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.展开更多
In this paper,an SIRS epidemic model using Grunwald-Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the pop...In this paper,an SIRS epidemic model using Grunwald-Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases.The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number.The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter.Additionally,using the Grunwald-Letnikov approximation,the simulation is carried out to confirm the validity of analytic results graphically.Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021,we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one.Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases.展开更多
This work investigates a prey-predator model featuring a Holling-type II functional response,in which the fear effect of predation on the prey species,as well as prey refuge,are considered.Specifically,the model assum...This work investigates a prey-predator model featuring a Holling-type II functional response,in which the fear effect of predation on the prey species,as well as prey refuge,are considered.Specifically,the model assumes that the growth rate of the prey population decreases as a result of the fear of predators.Moreover,the detection of the predator by the prey species is subject to a delay known as the fear response delay,which is incorporated into the model.The paper establishes the preliminary conditions for the solution of the delayed model,including positivity,boundedness and permanence.The paper discusses the existence and stability of equilibrium points in the model.In particular,the paper considers the discrete delay as a bifurcation parameter,demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter.The direction and stability of periodic solutions are determined using central manifold and normal form theory.Additionally,the global stability of the model is established at axial and positive equilibrium points.An extensive numerical simulation is presented to validate the analytical findings,including the continuation of the equilibrium branch for positive equilibrium points.展开更多
Population ecology theory is replete with density-dependent processes.However,traitmediated or behavioral indirect interactions can both reinforce or oppose densitydependent effects.This paper presents the first two s...Population ecology theory is replete with density-dependent processes.However,traitmediated or behavioral indirect interactions can both reinforce or oppose densitydependent effects.This paper presents the first two species competitive ODE and PDE systems,where the non-consumptive behavioral fear effect and the Allee effect,a densitydependent process,are both present.The stability of the equilibria is discussed analytically using the qualitative theory of ordinary differential equations.It is found that the Allee effect and the fear effect change the extinction dynamics of the system and the number of positive equilibrium points,but they do not affect the stability of the positive equilibria.We also observe standard co-dimension one bifurcation in the system by varying the Allee or fear parameter.Interestingly,we find that the Allee effect working in conjunction with the fear effect can bring about several dynamical changes to the system with only fear.There are three parametric regimes of interest in the fear parameter.For small and intermediate amounts of fear,the Allee+fear effect opposes dynamics driven by the fear effect.However,for large amounts of fear the Allee+fear effect reinforces the dynamics driven by the fear effect.The analysis of the corresponding spatially explicit model is also presented.To this end,the comparison principle for parabolic PDE is used.The conclusions of this paper have strong implications for conservation biology,biological control as well as the preservation of biodiversity.展开更多
In this paper,a delayed diffusive predator-prey model with fear effect under Neumann boundary conditions is considered.For the system without diffusion and delay,the conditions for the existence and local stability of...In this paper,a delayed diffusive predator-prey model with fear effect under Neumann boundary conditions is considered.For the system without diffusion and delay,the conditions for the existence and local stability of equilibria are obtained by analyzing the eigenvalues.Then,the instability induced by diffusion and delay-diffusion of the positive constant stationary solutions are discussed,respectively.Moreover,the regions of instability and pattern formation can be achieved with respect to diffusion and delay coefficients.Furthermore,the existence and direction of Hopf bifurcation and the properties of the homogeneous/nonhomogeneous bifurcated periodic solutions are driven by using the center manifold theorem and the normal form theory.Finally,some numerical simulations are carried out to verify the theoretical results.展开更多
A four-dimensional mathematical model is formulated to explore the fear effect exerted by large carnivore in the grassland ecosystem.The model depicts the interactions among herbage,domestic herbivore,wild herbivore a...A four-dimensional mathematical model is formulated to explore the fear effect exerted by large carnivore in the grassland ecosystem.The model depicts the interactions among herbage,domestic herbivore,wild herbivore and large carnivore,which incorporates both direct predation and anti-predator mechanisms.The dynamic properties of the model are analytically investigated,including the dissipativity of solutions,and the existence and stability of different equilibria.Some numerical simulations are also presented to exhibit rich dynamical behaviors,such as various types of bistabilities,periodic oscillation and chaotic oscillation.The study reveals that the appropriate level of fear factors can stabilize the system and increase the density of herbage and domestic herbivore.The fear effect plays an important role in maintaining the balance of the grassland ecosystem and promoting the economy of human society.展开更多
In this paper,the predator-prey model with strong Allee and fear effects is considered.The existence of the equilibria and their stability are established.Especially it is found that there is an interesting degenerate...In this paper,the predator-prey model with strong Allee and fear effects is considered.The existence of the equilibria and their stability are established.Especially it is found that there is an interesting degenerate point,which is a cusp point with codimension 2 or higher codimension,or an attracting(repelling)-type saddle-node,subject to different conditions.Then the Hopf bifurcation and its direction,the saddle-node bifurcation and the Bogdanov-Tankens bifurcation are further explored.Afterwards,with the help of the energy estimates and the Leray-Schauder degree,the nonexistence and existence of the nonconstant steady states of the model are presented.From the obtained results,we find that strong Allee effect will cause the per capita growth rate of prey species from negative to positive;both the fear and Allee effects could affect the existence of equilibria and bifurcations;meanwhile,the diffusion rates will affect the existence of the nonconstant steady states.展开更多
In this study,we consider a diffusive predator-prey model with multiple Allee effects induced by fear factors.We investigate the existence,boundedness and permanence of the solution of the system.We also discuss the e...In this study,we consider a diffusive predator-prey model with multiple Allee effects induced by fear factors.We investigate the existence,boundedness and permanence of the solution of the system.We also discuss the existence and non-existence of non-constant solutions.We derive sufficient conditions for spatially homogeneous(non-homogenous)Hopf bifurcation and steady state bifurcation.Theoretical and numerical simulations show that strong Allee effect and fear effect have great effect on the dynamics of system.展开更多
In this paper, we consider a predator-prey model with fear effect and square root functional response. We give the singularity of the origin and discuss the stability and Hopf bifurcation of the trivial equilibrium an...In this paper, we consider a predator-prey model with fear effect and square root functional response. We give the singularity of the origin and discuss the stability and Hopf bifurcation of the trivial equilibrium and the positive equilibrium. We show that the fear effect has no effect on prey density, but will lead to the decrease of predator populations.展开更多
In this paper,we propose a delayed prey-predator-scavenger system with fear effect and linear harvesting.First,we discuss the existence and stability of all possible equilibria.Next,we investigate the existence of Hop...In this paper,we propose a delayed prey-predator-scavenger system with fear effect and linear harvesting.First,we discuss the existence and stability of all possible equilibria.Next,we investigate the existence of Hopf bifurcation of the delayed system by regarding the gestation period of the scavenger as a bifurcation parameter.Furthermore,we obtain the direction of Hopf bifurcation and the stability of bifurcating periodic solutions by using the normal form theory and the central manifold theorem.In addition,we give the optimal harvesting strategy of the delayed system based on Pontryagin's maximum principle with delay.Finally,some numerical simulations are carried out to verify our theoretical results.展开更多
In this paper,a stage structure predator-prey model consisting of three nonlinear ordinary differential equations is proposed and analyzed.The prey populations are divided into two parts:juvenile prey and adult prey.F...In this paper,a stage structure predator-prey model consisting of three nonlinear ordinary differential equations is proposed and analyzed.The prey populations are divided into two parts:juvenile prey and adult prey.From extensive experimental data,it has been found that prey fear of predators can alter the physiological behavior of individual prey,and the fear effect reduces their reproductive rate and increases their mortality.In addition,we also consider the presence of constant ratio refuge in adult prey populations.Moreover,we consider the existence of intraspecific competition between adult prey species and predator species separately in our model and also introduce the gestation delay of predators to obtain a more realistic and natural eco-dynamic behaviors.We study the positivity and boundedness of the solution of the non-delayed system and analyze the existence of various equilibria and the stability of the system at these equilibria.Next by choosing the intra-specific competition coeficient of adult prey as bifurcation parameter,we demonstrate that Hopf bifurcation may occur near the positive equilibrium point.Then by taking the gestation delay as bifurcation parameter,the suficient conditions for the existence of Hopf bifurcation of the delayed system at the positive equilibrium point are given.And the direction of Hopf bifurcation and the stability of the periodic solution are analyzed by using the center manifold theorem and normal form theory.What's more,numerical experiments are performed to test the theoretical results obtained in this paper.展开更多
基金supported by the National Natural Science Foundation of China (No.12271261)the National Undergraduate Training Program for Innovation and Entrepreneurship (No.202310300044Z)。
文摘In this paper,we establish a delayed predator-prey model with nonlocal fear effect.Firstly,the existence,uniqueness,and persistence of solutions of the model are studied.Then,the local stability,Turing bifurcation,and Hopf bifurcation of the constant equilibrium state are analyzed by examining the characteristic equation.The global asymptotic stability of the positive equilibrium point is investigated using the Lyapunov function method.Finally,the correctness of the theoretical analysis results is verified through numerical simulations.
文摘In this paper, the dynamical behaviors of a modified Leslie-Gower predator-prey model incorporating fear effect and prey refuge are investigated. We delve into the construction of the model and its biological significance, with preliminary results encompassing positivity, boundedness, and persistence. The stability of the system’s boundary and positive equilibrium points is proven by calculating the real part of the eigenvalues of the Jacobian matrix. At the positive equilibrium point, we demonstrate that the system’s unique positive equilibrium is globally asymptotically stable by using the Dulac criterion. Furthermore, at this equilibrium point, we employ the Implicit Function Theorem to discuss how fear effects and prey refuges influence the population densities of both prey and predators. Finally, numerical simulations are conducted to validate the above-mentioned conclusions and explored the impact of Predator-taxis sensitivity αon dynamics of the system.
文摘In this manuscript, we have studied a fractional-order tri-trophic model with the help of Caputo operator. The total population is divided into three parts, namely prey, intermediate predator and top predator. In addition, the predator fear impact on prey population is suggested in this paper. Existence and uniqueness along with non-negativity and boundedness of the model system have been investigated. We have studied the local stability at all equilibrium points. Also, we have discussed global stability and Hopf bifurcation of our suggested model at interior equilibrium point. The Adam-Bashforth-Moulton approach is utilized to approximate the solution to the proposed model. With the help of MATLAB, we were able to conduct graphical demonstrations and numerical simulations.
基金supported by the Natural Science Foundation of Huai'an(HAB202162).
文摘In this research,we examine a predator–prey model in which nonlocal fear plays a role alongside delay in a reaction–diffusion framework.We integrate two delays into the model to account for the lag between when fear starts affecting the growth rate of prey and when it starts affecting the growth rate of the predator through feedback.The first step is to investigate local and global stability and bifurcations in the equilibrium states of the nondelayed model.We explore the Hopf bifurcation in the delayed model using the delay as the bifurcation parameter.Our theoretical findings are then backed up by certain simulations.It reveals how the system,depending on its level of anxiety and the time delays involved,displays a wide range of spatiotemporal patterns.
文摘In this paper, we consider the fear effect and gestation delay, and then establish a delayed predator-prey model with cannibalism. Firstly, we prove the well-posedness of the model. Secondly, the existence and stability of all equilibriums of the system are studied. Thirdly, the Hopf bifurcation at the coexistence equilibrium is investigated, and the conditions for the occurrence of Hopf bifurcation at the unique positive equilibrium point of the system with delay are determined. Finally, the numerical simulation results show that as the time delay increases, the equilibrium loses its stability, and the system has periodic solution.
基金Soumitra Pal is thankful to the Council of Scientific and Industrial Research(CSIR),Government of India for providing financial support in the form of senior research fellowship(File No.09/013(0915)/2019-EMR-I).
文摘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.
文摘In this paper,an SIRS epidemic model using Grunwald-Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases.The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number.The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter.Additionally,using the Grunwald-Letnikov approximation,the simulation is carried out to confirm the validity of analytic results graphically.Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021,we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one.Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases.
基金supported by MATRICS,Science Engineering Research Board,Government of India(MTR/2020/000477).
文摘This work investigates a prey-predator model featuring a Holling-type II functional response,in which the fear effect of predation on the prey species,as well as prey refuge,are considered.Specifically,the model assumes that the growth rate of the prey population decreases as a result of the fear of predators.Moreover,the detection of the predator by the prey species is subject to a delay known as the fear response delay,which is incorporated into the model.The paper establishes the preliminary conditions for the solution of the delayed model,including positivity,boundedness and permanence.The paper discusses the existence and stability of equilibrium points in the model.In particular,the paper considers the discrete delay as a bifurcation parameter,demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter.The direction and stability of periodic solutions are determined using central manifold and normal form theory.Additionally,the global stability of the model is established at axial and positive equilibrium points.An extensive numerical simulation is presented to validate the analytical findings,including the continuation of the equilibrium branch for positive equilibrium points.
文摘Population ecology theory is replete with density-dependent processes.However,traitmediated or behavioral indirect interactions can both reinforce or oppose densitydependent effects.This paper presents the first two species competitive ODE and PDE systems,where the non-consumptive behavioral fear effect and the Allee effect,a densitydependent process,are both present.The stability of the equilibria is discussed analytically using the qualitative theory of ordinary differential equations.It is found that the Allee effect and the fear effect change the extinction dynamics of the system and the number of positive equilibrium points,but they do not affect the stability of the positive equilibria.We also observe standard co-dimension one bifurcation in the system by varying the Allee or fear parameter.Interestingly,we find that the Allee effect working in conjunction with the fear effect can bring about several dynamical changes to the system with only fear.There are three parametric regimes of interest in the fear parameter.For small and intermediate amounts of fear,the Allee+fear effect opposes dynamics driven by the fear effect.However,for large amounts of fear the Allee+fear effect reinforces the dynamics driven by the fear effect.The analysis of the corresponding spatially explicit model is also presented.To this end,the comparison principle for parabolic PDE is used.The conclusions of this paper have strong implications for conservation biology,biological control as well as the preservation of biodiversity.
基金supported by the National Natural Science Foundation of China(Nos.12171135 and 11771115)the Natural Science Foundation of Hebei Province(Nos.A2020201021 and A2019201396)+1 种基金the Post Doctor Start-up Foundation of Zhejiang Normal University(No.ZC304021906)the Research Funding for High-Level Innovative Talents of Hebei University(No.801260201242).
文摘In this paper,a delayed diffusive predator-prey model with fear effect under Neumann boundary conditions is considered.For the system without diffusion and delay,the conditions for the existence and local stability of equilibria are obtained by analyzing the eigenvalues.Then,the instability induced by diffusion and delay-diffusion of the positive constant stationary solutions are discussed,respectively.Moreover,the regions of instability and pattern formation can be achieved with respect to diffusion and delay coefficients.Furthermore,the existence and direction of Hopf bifurcation and the properties of the homogeneous/nonhomogeneous bifurcated periodic solutions are driven by using the center manifold theorem and the normal form theory.Finally,some numerical simulations are carried out to verify the theoretical results.
基金Pingping Cong is funded by the China Scholarship Council(No.202106620028)Meng Fan is funded by the National Natural Science Foundation of China(No.12071068)Xingfu Zou is funded by the Natural Sciences and Engineering Research Council of Canada(No.RGPIN-2022-04744).
文摘A four-dimensional mathematical model is formulated to explore the fear effect exerted by large carnivore in the grassland ecosystem.The model depicts the interactions among herbage,domestic herbivore,wild herbivore and large carnivore,which incorporates both direct predation and anti-predator mechanisms.The dynamic properties of the model are analytically investigated,including the dissipativity of solutions,and the existence and stability of different equilibria.Some numerical simulations are also presented to exhibit rich dynamical behaviors,such as various types of bistabilities,periodic oscillation and chaotic oscillation.The study reveals that the appropriate level of fear factors can stabilize the system and increase the density of herbage and domestic herbivore.The fear effect plays an important role in maintaining the balance of the grassland ecosystem and promoting the economy of human society.
基金supported by the National Natural Science Foundation of China(Nos.11971032 and 12271143)the China Postdoctoral Science Foundation(No.2021M701118).
文摘In this paper,the predator-prey model with strong Allee and fear effects is considered.The existence of the equilibria and their stability are established.Especially it is found that there is an interesting degenerate point,which is a cusp point with codimension 2 or higher codimension,or an attracting(repelling)-type saddle-node,subject to different conditions.Then the Hopf bifurcation and its direction,the saddle-node bifurcation and the Bogdanov-Tankens bifurcation are further explored.Afterwards,with the help of the energy estimates and the Leray-Schauder degree,the nonexistence and existence of the nonconstant steady states of the model are presented.From the obtained results,we find that strong Allee effect will cause the per capita growth rate of prey species from negative to positive;both the fear and Allee effects could affect the existence of equilibria and bifurcations;meanwhile,the diffusion rates will affect the existence of the nonconstant steady states.
基金supported by Special Foundation for Excellent Young Teachers and Principals Program of Jiangsu Province,China.
文摘In this study,we consider a diffusive predator-prey model with multiple Allee effects induced by fear factors.We investigate the existence,boundedness and permanence of the solution of the system.We also discuss the existence and non-existence of non-constant solutions.We derive sufficient conditions for spatially homogeneous(non-homogenous)Hopf bifurcation and steady state bifurcation.Theoretical and numerical simulations show that strong Allee effect and fear effect have great effect on the dynamics of system.
文摘In this paper, we consider a predator-prey model with fear effect and square root functional response. We give the singularity of the origin and discuss the stability and Hopf bifurcation of the trivial equilibrium and the positive equilibrium. We show that the fear effect has no effect on prey density, but will lead to the decrease of predator populations.
基金This work is supported by the National Natural Science Foundation of China(Grant Nos.11661050 and 11861044)the HongLiu First-class Disciplines Development Program of Lanzhou University of Technology.
文摘In this paper,we propose a delayed prey-predator-scavenger system with fear effect and linear harvesting.First,we discuss the existence and stability of all possible equilibria.Next,we investigate the existence of Hopf bifurcation of the delayed system by regarding the gestation period of the scavenger as a bifurcation parameter.Furthermore,we obtain the direction of Hopf bifurcation and the stability of bifurcating periodic solutions by using the normal form theory and the central manifold theorem.In addition,we give the optimal harvesting strategy of the delayed system based on Pontryagin's maximum principle with delay.Finally,some numerical simulations are carried out to verify our theoretical results.
基金supported by the Natural Science Foundation of China(11672074)the Natural Science Foundation of Fujian Province(2022J01192).
文摘In this paper,a stage structure predator-prey model consisting of three nonlinear ordinary differential equations is proposed and analyzed.The prey populations are divided into two parts:juvenile prey and adult prey.From extensive experimental data,it has been found that prey fear of predators can alter the physiological behavior of individual prey,and the fear effect reduces their reproductive rate and increases their mortality.In addition,we also consider the presence of constant ratio refuge in adult prey populations.Moreover,we consider the existence of intraspecific competition between adult prey species and predator species separately in our model and also introduce the gestation delay of predators to obtain a more realistic and natural eco-dynamic behaviors.We study the positivity and boundedness of the solution of the non-delayed system and analyze the existence of various equilibria and the stability of the system at these equilibria.Next by choosing the intra-specific competition coeficient of adult prey as bifurcation parameter,we demonstrate that Hopf bifurcation may occur near the positive equilibrium point.Then by taking the gestation delay as bifurcation parameter,the suficient conditions for the existence of Hopf bifurcation of the delayed system at the positive equilibrium point are given.And the direction of Hopf bifurcation and the stability of the periodic solution are analyzed by using the center manifold theorem and normal form theory.What's more,numerical experiments are performed to test the theoretical results obtained in this paper.
