Maximal total population of species in a diffusive logistic model

Area of Science:

• Mathematical Biology
• Population Dynamics
• Differential Equations

Background:

• Investigating population dynamics governed by diffusive logistic equations.
• Understanding resource distribution effects on species' total population.
• Exploring qualitative properties of population maximizers under diffusion.

Purpose of the Study:

• To maximize the total population of a single species using a stationary diffusive logistic equation.
• To analyze the impact of large diffusivity on population maximizers.
• To derive an optimality condition and develop a numerical algorithm for computing the population maximizer.

Main Methods:

• Analysis of stationary diffusive logistic equations.
• Application of rearrangement theory to derive optimality conditions.
• Development and convergence analysis of a numerical algorithm for maximizer computation.
• Numerical simulations on various domains.

Main Results:

• Confirmed that concentrated resources are favorable for maximizing total population with large diffusivity.
• Derived an optimality condition for the population maximizer.
• Developed and validated a convergent numerical algorithm for computing the maximizer.
• Gained insights into maximizer properties through numerical simulations.

Conclusions:

• Concentrated resources are beneficial for population maximization in diffusive logistic models, especially with high diffusivity.
• The developed numerical algorithm is efficient and convergent across different geometries.
• Numerical simulations provide valuable insights and suggest further research directions/conjectures.