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
This paper deals with non-autonomous fractional stochastic reaction-diffusion equations driven by multiplicative noise with s∈(0,1).We first present some conditions for estimating the boundedness of fractal dimension...This paper deals with non-autonomous fractional stochastic reaction-diffusion equations driven by multiplicative noise with s∈(0,1).We first present some conditions for estimating the boundedness of fractal dimension of a random invariant set.Then we establish the existence and uniqueness of tempered pullback random attractors.Finally,the finiteness of fractal dimension of the random attractors is proved.展开更多
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The authors would like to thank the reviewers for their helpful comments.This work was partially supported by the National Natural Science Foundation of China(11871138)joint research project of Laurent Mathematics Center of Sichuan Normal UniversityNational-Local Joint Engineering Laboratory of System Credibility Automatic Verification,funding of V.C.&V.R.Key Lab of Sichuan Province.
文摘This paper deals with non-autonomous fractional stochastic reaction-diffusion equations driven by multiplicative noise with s∈(0,1).We first present some conditions for estimating the boundedness of fractal dimension of a random invariant set.Then we establish the existence and uniqueness of tempered pullback random attractors.Finally,the finiteness of fractal dimension of the random attractors is proved.
