Dynamics of a Predator-Prey System with Asymmetric Dispersal and Fear Effect
Abstract
:1. Introduction
2. Mathematical Formulation
- (a)
- We first consider and as the prey populations in the source and sink patches, respectively, while y represents the predator population.
- (b)
- We further assume that the predator only captures the prey population in the source patch and intraspecific competition only among the population of the prey in the source patch.
- (c)
- It is assumed that the prey population can disperse between the sink patch and the source patch.
- (a)
- which implies that if the prey does not have fear effect or the predator does not exist, then there is no reduction in the birth and mortality rates of the prey.
- (b)
- which means that if the level of fear effect or the density of predator is extremely high, the birth rate of the prey will eventually approach to zero.
- (c)
- which means that as the level of fear effect or the density of the predator increase, the birth rate of the prey decreases.
- (d)
- which means that if the level of fear effect is large or the density of the predator is high, then the death rate of prey will reach a maximum value.
- (e)
- which means that as either the fear level or the predator population increases, the mortality rate of the prey rises.So, incorporating all of the above facts, in this paper, we propose the following system:
3. Preliminaries
3.1. Non-Negativity and Boundedness of the Solutions
3.2. Permanence of the System (6)
4. Existence and Local Stability of the Equillbria
- (a)
- it always has a trivial equilibrium ;
- (b)
- it has a boundary equilibrium when ;
- (c)
- it has a unique positive equilibrium point if condition holds.
- (a)
- locally asymptotically stable when holds;
- (b)
- unstable when holds;
- (c)
- a saddle-node when .
- (a)
- a locally asymptotically stable node when holds;
- (b)
- a saddle when holds;
- (c)
- a saddle-node when
5. Global Stability Analysis
6. Bifurcation Analysis
7. Numerical Simulations
- Case I. Global stability of the equilibria
- Case II. Total population abundance
- Case III. Impact of fear effect on the system (6)
8. Conclusions
- We first investigated the non-negativity and boundedness of system (6), and proved that system (6) is persistent if . Furthermore, our research indicated that system (6) always has a trivial equilibrium point , and in the case of , there exists a boundary equilibrium point . Additionally, if condition holds, the system also possesses a unique positive equilibrium point .
- When the parameter reaches or exceeds the critical value , the equilibrium becomes globally asymptotically stable. Conversely, when and , the boundary equilibrium point is globally asymptotically stable. By applying geometric methods, we further investigated the global asymptotic stability of the system around the equilibrium point .
- When the parameter reaches the critical value , the boundary equilibrium point coincides with the trivial equilibrium point , and a transcritical bifurcation occurs near with a switch in stability. As the dispersal rate approaches or exceeds the critical value , the migration of prey between habitat patches becomes extremely sensitive, potentially leading to species collapse. This finding provides crucial guidance for species conservation and habitat connectivity management. In the construction of ecological corridors, it is essential to ensure their effectiveness and prevent excessive dispersal of prey from source patches to sink patches, which is vital for preventing species extinction.
- Compared to the no-dispersal scenario, there is an optimal dispersal coefficient at which the prey population reaches its maximum. However, when exceeds , the prey population begins to decline gradually. Upon reaching , the prey population returns to the no-dispersal level, but as continues to increase, the prey population begins to fall below that of the no-dispersal case, eventually leading to extinction (see Figure 4).
- Next, we specifically investigated the impact of the fear effect parameter k and the maximum fear cost on the dynamics of system (6). Next, we specifically examined the effects of the fear effect parameter k and the maximum fear cost on the dynamics of system (6). Consistent with [42], our findings indicate that the fear effect reduces predator population density but does not alter the existence or stability of equilibrium points. However, with the introduction of maximum fear effects in our model, we further observed that higher maximum fear levels not only suppress predator density more significantly but may also accelerate population decline, potentially impacting the long-term stability of the ecosystem. From Figure 5 and Figure 6, it is evident that k and do not significantly affect the density of the prey but they lead to a continuous decrease in the final stable state of the predator.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Equilibrium | Existence | Stability |
---|---|---|
always | L.A.S. | |
Saddle-node | ||
Unstable | ||
L.A.S. | ||
Saddle-node | ||
Unstable | ||
L.A.S. |
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Meng, X.; Chen, L.; Chen, F. Dynamics of a Predator-Prey System with Asymmetric Dispersal and Fear Effect. Symmetry 2025, 17, 329. https://doi.org/10.3390/sym17030329
Meng X, Chen L, Chen F. Dynamics of a Predator-Prey System with Asymmetric Dispersal and Fear Effect. Symmetry. 2025; 17(3):329. https://doi.org/10.3390/sym17030329
Chicago/Turabian StyleMeng, Xinyu, Lijuan Chen, and Fengde Chen. 2025. "Dynamics of a Predator-Prey System with Asymmetric Dispersal and Fear Effect" Symmetry 17, no. 3: 329. https://doi.org/10.3390/sym17030329
APA StyleMeng, X., Chen, L., & Chen, F. (2025). Dynamics of a Predator-Prey System with Asymmetric Dispersal and Fear Effect. Symmetry, 17(3), 329. https://doi.org/10.3390/sym17030329