Finite difference approximations for a size-structured population model with distributed states in the recruitment

Area of Science:

• Mathematical Biology
• Numerical Analysis
• Population Dynamics

Background:

• Size-structured population models are crucial for understanding population dynamics.
• Recruitment at different sizes presents unique modeling challenges.
• Classical models often assume a single recruitment size.

Purpose of the Study:

• To develop and analyze finite difference schemes for a size-structured population model with distributed recruitment.
• To prove convergence of these schemes to a unique weak solution.
• To investigate the model's behavior, including convergence to classical models and the occurrence of Hopf-bifurcation.

Main Methods:

• Development of first- and second-order finite difference schemes.
• Mathematical proof of convergence to a unique weak solution.
• Analysis of the weak* topology convergence to Gurtin-McCamy-type models.
• Numerical simulations to validate accuracy and performance.
• Application of schemes to demonstrate Hopf-bifurcation.

Main Results:

• Convergence of finite difference approximations to a unique weak solution is proven.
• The weak solution of the distributed model converges to the classical Gurtin-McCamy model as recruit size distribution concentrates.
• The second-order scheme shows superior performance for discontinuities.
• Supercritical Hopf-bifurcation is demonstrated in the distributed model using the second-order scheme.

Conclusions:

• The developed finite difference schemes are accurate and convergent for size-structured population models with distributed recruitment.
• The distributed recruitment model approximates classical models under specific conditions.
• The study confirms the presence of Hopf-bifurcation in distributed recruitment models, highlighting the importance of considering recruitment size distributions.