[Paper Review] On competition through growth reduction

Carles Barril, Àngel Calsina+2 more|arXiv (Cornell University)|Mar 6, 2023

Mathematical and Theoretical Epidemiology and Ecology ModelsMedicine3 citations

TL;DR

This paper studies a size-structured population model where individuals grow at a rate reduced by taller conspecifics, modeling competition via light shading. Using a scalar renewal equation and delay differential equations, it proves the existence and stability of a unique non-trivial stationary birth rate under general conditions, linking results to a quasilinear PDE formulation and establishing stability via linearised analysis.

ABSTRACT

We consider a population organised hierarchically with respect to size in such a way that the growth rate of each individual depends only on the presence of larger individuals. As a concrete example one might think of a forest, in which the incidence of light on a tree (and hence how fast it grows) is affected by shading of taller trees. The model is formulated as a delay equation, more specifically a scalar renewal equation, for the population birth rate. After discussing the well-posedness of the model, we analyse how many stationary birth rates the equation can have in terms of the functional parameters of the model. In particular we show that, under reasonable and rather general assumptions, only one stationary birth rate can exist besides the trivial one (associated to the state in which there are no individuals and the population birth rate is zero). We give conditions for this non-trivial stationary birth rate to exist and we analyse its stability using the principle of linearised stability for delay equations. Finally we relate the results to an alternative formulation of the model taking the form of a quasilinear partial differential equation for the population size-density.

Motivation & Objective

• To model population dynamics where growth is reduced by larger individuals, motivated by light competition in forests.
• To analyze the existence and stability of non-zero stationary birth rates in such a system.
• To establish a correspondence between the delay equation formulation and a quasilinear PDE model of population density.
• To apply linearised stability theory to determine conditions under which the non-trivial steady state is locally asymptotically stable.

Proposed method

• Formulates the model as a scalar nonlinear renewal equation for the population birth rate, with growth rate dependent on the number of taller individuals.
• Introduces an interaction variable $ E(x,t) = \int_x^\infty u(s,t)\,ds $, representing the cumulative shading effect from individuals taller than $ x $.
• Analyzes the system using delay differential equations, deriving a characteristic equation for linearised stability around steady states.
• Applies the principle of linearised stability for delay equations to determine stability of the non-trivial stationary birth rate.
• Derives an equivalent quasilinear first-order PDE formulation for the population size-density, incorporating non-local feedback through $ E(x,t) $.
• Establishes equivalence between conditions for non-trivial stationary densities in the PDE and non-trivial stationary birth rates in the delay formulation.

Experimental results

Research questions

• RQ1 Under what conditions does a non-trivial stationary birth rate exist in a population where growth is reduced by larger individuals?
• RQ2 How does the stability of the trivial (zero) birth rate depend on the net reproduction number $ R(0) $?
• RQ3 What is the relationship between the delay equation formulation and the quasilinear PDE formulation of the same population model?
• RQ4 Can the stability of the non-trivial steady state be determined via linearised analysis of the delay equation?
• RQ5 How do the functional parameters—fertility $ \beta(x) $, growth rate $ g(E) $, and death rate $ \mu $—affect the existence and stability of equilibria?

Key findings

• Under general and reasonable assumptions, the model admits at most one non-trivial stationary birth rate in addition to the trivial zero solution.
• A non-trivial stationary birth rate exists if and only if the net reproduction number at zero population, $ R(0) = \int_{x_m}^\infty \frac{\beta(x)}{g(0)} e^{-\frac{\mu}{g(0)}(x-x_m)}\,dx $, exceeds one.
• The trivial steady state is globally asymptotically stable when it is the only stationary birth rate, i.e., when $ R(0) < 1 $.
• For a two-parameter family of fertility functions, the non-trivial stationary birth rate is locally asymptotically stable when it exists.
• The characteristic equation for linearised stability in the delay formulation matches that derived from the PDE formulation, confirming consistency between the two models.
• The stability of the non-trivial steady state can be inferred from the spectral properties of the linearised operator, with the leading eigenvalue determining stability.