Adaptation in a heterogeneous environment II: to be three or not to be

Area of Science:

• Mathematical Biology
• Population Dynamics
• Ecological Modeling

Background:

• Phenotypic adaptation in structured populations is crucial for ecological and evolutionary processes.
• Understanding population persistence in heterogeneous environments is vital for conservation and disease management.

Purpose of the Study:

• To develop and analyze a mathematical model for phenotypically structured populations in a multi-patch environment.
• To investigate how the number and characteristics of patches influence population persistence.

Main Methods:

• Rigorous mathematical analysis including fixed point theorems, comparison principles, and integral estimates.
• Exploration of eigenvalue dependency on the number of patches (H).
• Numerical simulations to validate theoretical findings.

Main Results:

• Population persistence is determined by the sign of a principal eigenvalue.
• The impact of adding patches (springboard or detrimental effect) is complex and depends on the distribution of optimal phenotypes.
• A detailed characterization of the conditions under which a third patch alters persistence probability compared to two patches is provided.

Conclusions:

• The study provides novel insights into the complex relationship between environmental heterogeneity and population persistence.
• The findings have direct implications for agroecology and the study of zoonotic diseases by modeling pathogen adaptation across host species.
• The mathematical framework developed offers a powerful tool for analyzing similar ecological problems with asymmetric patch structures.