Stability and pattern formation for competing populations with asymmetric nonlocal coupling.

Area of Science:

• Ecology
• Mathematical Biology
• Theoretical Ecology

Background:

• Ecological models often simplify species interactions to local competition.
• Nonlocal interactions, accounting for species mobility, are crucial for understanding complex population dynamics.
• Asymmetric nonlocal coupling introduces directional influences in species interactions.

Purpose of the Study:

• To investigate the impact of asymmetric nonlocal coupling on competing species models.
• To analyze the stability of coexistence equilibria under varying degrees of nonlocality and asymmetry.
• To explore emergent spatial-temporal patterns in population dynamics.

Main Methods:

• Development of a two-species competition model with asymmetric nonlocal coupling via convolution integrals.
• Linear stability analysis to determine critical parameters for destabilization.
• Numerical computation of nonlinear patterns and stability boundaries using asymmetric Gaussian and step functions as kernels.

Main Results:

• Stable coexistence equilibria can be destabilized by increasing nonlocality (species mobility).
• Emergent patterns include arrays of population islands separated by deadzones.
• Observed dynamics include propagating traveling waves, colony formation, and modulated traveling waves, explained by positive feedback mechanisms.

Conclusions:

• Asymmetric nonlocal coupling significantly alters population dynamics, leading to pattern formation.
• Species mobility and interaction asymmetry are key drivers of complex ecological structures.
• The model provides insights into mechanisms underlying phenomena like colony formation and wave propagation in ecological systems.