Generalizing Fisher's "reproductive value": linear differential and difference equations of "dilute" biological...

Area of Science:

• Population Dynamics
• Mathematical Biology
• Ecology

Background:

• R. A. Fisher's 1930 concept of reproductive value defines initial population contributions to exponential growth.
• Reproductive value is a characteristic row vector in the Leslie matrix and an eigenfunction in Lotka's integral equation model.
• It extends to 2-sex models and general n-variable linear systems.

Purpose of the Study:

• To generalize the concept of reproductive value to more complex linear systems.
• To prepare for extending reproductive value to nonlinear population dynamics.
• To investigate reproductive value under resource limitations where linearity breaks down.

Main Methods:

• Analysis of linear population models, including discrete-time Leslie models and Lotka's integral equation model.
• Generalization of reproductive value to n-variable linear systems and 2-sex models.
• Exploration of the loss of positive definability in reproductive value when resource limitations induce nonlinearity.

Main Results:

• Reproductive value is generalized to various linear population models.
• The study identifies limitations of reproductive value in nonlinear systems due to resource constraints.
• A foundation is established for future research on reproductive value in nonlinear contexts.

Conclusions:

• The generalization of reproductive value to linear systems is robust.
• Resource limitations fundamentally alter reproductive value, necessitating nonlinear approaches.
• Further research is needed to define reproductive value for nonlinear population dynamics.