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Stability analysis and long-time convergence of a partial differential equation model of two-phase ageing

Luce Breuil

Abstract

Recent biological evidence suggests the presence of a two-phase ageing process in several species. We introduce a system of two age-structured partial differential equations (PDE) representing two phases of ageing of a wild population. The model includes a coupling of both equations through birth and transition between phases and non-linearities due to competition. We show the existence, positivity and uniqueness of weak solutions in a general setting. For a simplified system of ordinary differential equations (ODE), we show existence and uniqueness of a strictly positive steady state attracting all trajectories. We study another simplification, a coupled PDE-ODE model, for which we prove existence, uniqueness and local asymptotic stability of a strictly positive steady state. Under further assumptions, but without assuming weak non-linearities, we show the global asymptotic stability of that steady state. The uniqueness of steady states and absence of oscillations in these systems show that the proportion of individuals in each phase at equilibrium is a unique feature of the model. This paves the way to ecological applications as the experimental measure of such a proportion could help gain some insight on the health of a wild population.

Stability analysis and long-time convergence of a partial differential equation model of two-phase ageing

Abstract

Recent biological evidence suggests the presence of a two-phase ageing process in several species. We introduce a system of two age-structured partial differential equations (PDE) representing two phases of ageing of a wild population. The model includes a coupling of both equations through birth and transition between phases and non-linearities due to competition. We show the existence, positivity and uniqueness of weak solutions in a general setting. For a simplified system of ordinary differential equations (ODE), we show existence and uniqueness of a strictly positive steady state attracting all trajectories. We study another simplification, a coupled PDE-ODE model, for which we prove existence, uniqueness and local asymptotic stability of a strictly positive steady state. Under further assumptions, but without assuming weak non-linearities, we show the global asymptotic stability of that steady state. The uniqueness of steady states and absence of oscillations in these systems show that the proportion of individuals in each phase at equilibrium is a unique feature of the model. This paves the way to ecological applications as the experimental measure of such a proportion could help gain some insight on the health of a wild population.
Paper Structure (19 sections, 28 theorems, 195 equations, 3 figures)

This paper contains 19 sections, 28 theorems, 195 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Presentation of the model and existence of solutions
  3. Presentation of the model
  4. Eigenvalue problem for the linearised system
  5. Study of the linear system
  6. Existence and uniqueness of solutions in the general case
  7. Characterisation of steady states
  8. Long-time asymptotics in the ODE case
  9. Local and global stability for a PDE-ODE model
  10. Steady state analysis
  11. Local stability
  12. Global long-time convergence
  13. Appendix
  14. Proofs of Section \ref{['sec:gen_lin']}
  15. Proofs of Section \ref{['sec:ex_un']}
  16. ...and 4 more sections

Key Result

Proposition 2.1

Suppose Assumptions assump:bio and ass:for_lambda_0 are verified. Then, systems eq:nns_0 and eq:phi_0 have unique solutions with $\lambda_0 >0$ and the solutions to eq:phi_0 are positive. Provided Assumption assump:boundsbound:b is also verified, the solutions to eq:phi_0 are bounded. Finally, if $b

Figures (3)

  • Figure 1: Schematic representation of the model.
  • Figure 2: Case $A > 0$.
  • Figure 3: Case $A \leq 0$

Theorems & Definitions (61)

  • Proposition 2.1
  • proof
  • Definition 2.1: Weak solution
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2: Long time convergence
  • Theorem 2.3
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • ...and 51 more