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Instant cost and delayed reward. Demographic eco-evolutionary game dynamics under the impact of the delay resulting from the offspring maturation time

Krzysztof Argasiński, Ryszard Rudnicki, Robert Szczelina

TL;DR

The paper investigates how maturation delay of juveniles, incorporated as a delay in fertility payoffs, reshapes demographic eco-evolutionary game dynamics using Delay Differential Equations. It decomposes reproduction and survival into explicit birth and death processes, derives a general framework for fertility-delayed replicator dynamics, and develops analytical and numerical tools to identify critical delays and complex dynamical regimes. Three juvenile-survival modes—no delay, logistic suppression, and delayed suppression—reveal a spectrum of behaviors from stable equilibria to bifurcations, cycles, and chaos, with especially rich dynamics when delay is tied to juvenile recruitment at birth. The work highlights ghost attractors and long transients that intensify with delay, emphasizing the strong influence of life-history timing and density dependence on eco-evolutionary outcomes and resilience to perturbations.

Abstract

In this paper, we extend the demographic eco-evolutionary game approach, based on explicit birth and death dynamics instead of abstract "fitness" interpreted as an abstract "Malthusian parameter", by the introduction of the delay resulting from the juvenile maturation time. This leads to the application of the Delay Differential Equations (DDE). We show that delay seriously affects the resulting dynamics and may lead to the loss of stability of equilibria when critical delay is exceeded. We provide theoretical tools for the assessment of the critical delays and the parameter values when this may happen. Our results emphasize the importance of the mechanisms of density dependence. We analyze the impact of three different suppression modes based on: adult mortality, juvenile recruitment survival after the maturation period (without delay), and juvenile recruitment at birth (with the delay). The last mode leads to extreme patterns such as bifurcations, complex cycles, and chaotic dynamics. However, surprisingly, this mode leads to extension of the duration of the temporary transient metastable states known as "ghost attractors". In addition, we also focus on the problem of resilience of the analyzed systems against external periodic perturbations and feedback-driven factors such as additional predator pressure.

Instant cost and delayed reward. Demographic eco-evolutionary game dynamics under the impact of the delay resulting from the offspring maturation time

TL;DR

The paper investigates how maturation delay of juveniles, incorporated as a delay in fertility payoffs, reshapes demographic eco-evolutionary game dynamics using Delay Differential Equations. It decomposes reproduction and survival into explicit birth and death processes, derives a general framework for fertility-delayed replicator dynamics, and develops analytical and numerical tools to identify critical delays and complex dynamical regimes. Three juvenile-survival modes—no delay, logistic suppression, and delayed suppression—reveal a spectrum of behaviors from stable equilibria to bifurcations, cycles, and chaos, with especially rich dynamics when delay is tied to juvenile recruitment at birth. The work highlights ghost attractors and long transients that intensify with delay, emphasizing the strong influence of life-history timing and density dependence on eco-evolutionary outcomes and resilience to perturbations.

Abstract

In this paper, we extend the demographic eco-evolutionary game approach, based on explicit birth and death dynamics instead of abstract "fitness" interpreted as an abstract "Malthusian parameter", by the introduction of the delay resulting from the juvenile maturation time. This leads to the application of the Delay Differential Equations (DDE). We show that delay seriously affects the resulting dynamics and may lead to the loss of stability of equilibria when critical delay is exceeded. We provide theoretical tools for the assessment of the critical delays and the parameter values when this may happen. Our results emphasize the importance of the mechanisms of density dependence. We analyze the impact of three different suppression modes based on: adult mortality, juvenile recruitment survival after the maturation period (without delay), and juvenile recruitment at birth (with the delay). The last mode leads to extreme patterns such as bifurcations, complex cycles, and chaotic dynamics. However, surprisingly, this mode leads to extension of the duration of the temporary transient metastable states known as "ghost attractors". In addition, we also focus on the problem of resilience of the analyzed systems against external periodic perturbations and feedback-driven factors such as additional predator pressure.
Paper Structure (24 sections, 1 theorem, 91 equations, 31 figures)

This paper contains 24 sections, 1 theorem, 91 equations, 31 figures.

Table of Contents

  1. Introduction
  2. Methods and technical details from previous papers:
  3. Resilience against seasonal perturbations and external feedback driven by predator pressure
  4. Hawk-Dove example without delays
  5. Results
  6. Abstract "fitness" or explicit births and deaths? Interaction rates, pair formation and the timing of interaction events
  7. Meaning and interpretation of the time unit
  8. Methods used in analysis of the impact of the delays
  9. Analytical toolbox
  10. Numerical toolbox
  11. Derivation of the general framework for replicator dynamics with fertility delay resulting from the egg hatching or maturation time.
  12. Density dependence
  13. Hawk-Dove game example
  14. Numerical simulations
  15. Case without juvenile recruitment mortality, type a) juvenile survival
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\mathbf{A}$ be the matrix of the form $\mathbf{A} =[a_{ij}]$, $1\leq i,j\leq 2$. Assume that the rest point $(x^{\ast },y^{\ast })$ of system (eq-no)--(eq-n1o) is locally stable for ${ \if@compatibility \mathchar"010D {} \mathchar"010D } =0$. Let and Let $u>0$ be a constant which satisfies the equation The stationary solution loses its stability (${ \if@compatibility \mathchar"010

Figures (31)

  • Figure 1: Case without juvenile mortality factors.Trajectory for parameters $F=0.9$, $d=1$, $\Phi =0.5$, $\Psi =0.2$ and $\Omega =0.0001$ and the constant initial history $q_{d}=0.8$ and $n=100$.
  • Figure 2: Case with non-constant initial history. Initial frequency linearly grows from $0.35$ to $0.8$ and the population size declines linearly from $3100$ to $100$.
  • Figure 3: Trajectories with added periodic background mortality with amplitude ${ \if@compatibility \mathchar"010B {} \mathchar"010B } =0.2$ and period ${ \if@compatibility \mathchar"0112 {} \mathchar"0112 } =120$.
  • Figure 4: Case with added predator pressure. Parameters of the predator-prey subsystem are $b_{p}=0.3$ ,$\ d_{p}=0.8$ and $p=0.6$.
  • Figure 5: Phase portraits of the system from Fig.4
  • ...and 26 more figures

Theorems & Definitions (1)

  • Theorem 1