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ABSTRACT Predator‐prey models, such as the Leslie‐Gower model, are essential for understanding population dynamics and stability within ecosystems. These models help explain the balance between species under natural conditions, but the inclusion of factors like the Allee effect and intraspecific competition adds complexity and realism to these interactions, enhancing our ability to predict system behavior under stress. To detect early indicators of population collapse, this study investigates the intricate dynamics of a modified Leslie‐Gower predator‐prey model with both Allee effect and intraspecific competition. We analyze the existence and stability of equilibria, as well as bifurcation phenomena, including saddle‐node bifurcations of codimension 2, Hopf bifurcations of codimension 2, and Bogdanov‐Takens bifurcations of codimension at least 4. Detailed transitions between bifurcation curves–specifically saddle‐node, Hopf, homoclinic, and limit cycle bifurcations–are also examined. We observe a novel transition phenomenon, where a system jumps from saddle‐node bifurcation to homoclinic and limit cycle bifurcations. This suggests that burst oscillations may serve as an early warning of system collapse rather than simply a tipping point. Our findings indicate that moderate levels of intraspecific competition or Allee effect support coexistence of both populations, while excessive levels may destabilize the entire biological system, leading to collapse. These insights offer valuable implications for ecological management and the early detection of risks in population dynamics.
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Dynamics of a linearly perturbed May–Leonard competition model
The May–Leonard model was introduced to examine the behavior of three competing populations where rich dynamics, such as limit cycles and nonperiodic cyclic solutions, arise. In this work, we perturb the system by adding the capability of global mutations, allowing one species to evolve to the other two in a linear manner. We find that for small mutation rates, the perturbed system not only retains some of the dynamics seen in the classical model, such as the three-species equal-population equilibrium bifurcating to a limit cycle, but also exhibits new behavior. For instance, we capture curves of fold bifurcations where pairs of equilibria emerge and then coalesce. As a result, we uncover parameter regimes with new types of stable fixed points that are distinct from the single- and dual-population equilibria characteristic of the original model. On the contrary, the linearly perturbed system fails to maintain heteroclinic connections that exist in the original system. In short, a linear perturbation proves to be significant enough to substantially influence the dynamics, even with small mutation rates.
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- Award ID(s):
- 2151566
- PAR ID:
- 10426089
- Date Published:
- Journal Name:
- Chaos: An Interdisciplinary Journal of Nonlinear Science
- Volume:
- 33
- Issue:
- 6
- ISSN:
- 1054-1500
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0138150
