Nonlinear semelparous leslie models.

Area of Science:

• Mathematical Biology
• Population Dynamics
• Nonlinear Dynamics

Background:

• Leslie matrix models are fundamental for understanding structured population dynamics.
• Analysis of bifurcations at trivial equilibria is crucial for predicting population persistence.
• Previous models often assume generic bifurcations, limiting applicability to specific scenarios.

Purpose of the Study:

• To investigate bifurcations at the trivial equilibrium in a general class of nonlinear Leslie matrix models.
• To analyze the behavior of population dynamics when only the oldest age class reproduces.
• To determine the conditions under which population equilibria or cycles emerge and their stability.

Main Methods:

• Utilized nonlinear Leslie matrix models with a focus on the net reproductive number (n) as a bifurcation parameter.
• Analyzed the global branch of positive equilibria and single-class cycles bifurcating from the trivial equilibrium at n = 1.
• Investigated the stability of bifurcating solutions based on model parameters, including inter- and intra-class competition.

Main Results:

• A global branch of positive equilibria bifurcates from the trivial equilibrium at n = 1, even for nongeneric bifurcations.
• Bifurcating equilibria can be supercritical or subcritical, with stability not solely determined by bifurcation direction.
• Single-class cycles also bifurcate at n = 1; in two-class models, either equilibria or cycles are stable, influenced by competition strength, leading to temporally separated populations with strong inter-class competition.

Conclusions:

• Nonlinear Leslie matrix models exhibit complex bifurcations at n = 1, leading to diverse population dynamics.
• The stability of population structures (equilibria vs. cycles) is contingent on inter- and intra-class competition.
• For multi-class populations, bifurcating cycles can form invariant loops, potentially acting as attractors under specific conditions.