Adding a fecundity-survival trade-off to a discrete population model with maturation delay

TL;DR

The paper develops discrete-delay population models that incorporate a maturation delay with a fecundity–survival trade-off, linking body-size–driven fecundity via $g(\tau)=a-be^{-K(\tau+1)}$ to survival structures for mature and immature cohorts. It analyzes four model variants formed by pairing Beverton--Holt or Ricker survival for adults with either constant or cohort-density-dependent immature survival, showing an emergent positive optimal maturation delay that maximizes equilibrium abundance and a critical delay threshold leading to extinction. In Ricker-based cases, longer delays can induce oscillations (period-doubling or Neimark--Sacker) before potential extinction, revealing rich dynamical regimes driven by maturation delay and density-dependence. Overall, the framework provides a versatile phenomenological tool to study life-history trade-offs across parental and offspring survival and reproductive investment, with clear implications for optimal maturation strategies and extinction risk.

Abstract

Although maturation delays are frequently included in population models, researchers rarely account for mortality between birth and maturity. Previous discrete population models have included mortality of immature individuals during the maturation delay finding that increasing the delay decreases the equilibrium population size, eventually leading to extinction. Since maturation delays beyond one breeding cycle are often found in nature, they must also have a benefit leading to a trade-off. We derive a class of models to explore the trade-off between the benefit of a longer maturation delay on fecundity due to larger body sizes at maturity and the down-side on survival. We examine two scenarios: density independent survival and cohort density dependent survival of immature individuals. For the mature and immature individuals, we consider two different, but popular, survival functions: the Beverton--Holt model and the Ricker model. Across all models, we identify a positive maturation delay that maximizes the population size that we refer to as the ``optimal maturation delay'' and a critical delay threshold that results in extinction. We also find oscillatory dynamics with the Ricker survival function for certain ranges of maturation delay. Overall, our delay model sets up a useful phenomenological framework to test multiple combinations of trade-offs in parent survival, offspring survival, and reproductive investment.

Figures (10)

• Figure 1: Dependence of $N^*$ on $\tau \in \mathbb{N}^+_0$ for different values of $\overline{p}\in (0,1)$. Parameter values are $K=1, b=200, a=10, \alpha=0.1, \beta=0.3$. $N^*(\tau)=\max\{N^*_0=0,N^*_+(\tau)\}$.
• Figure 2: a) Bifurcation diagram for the Ricker-Constant model with $\tau=2,4,6$. Vertical lines with a star and circle denote the $a$ values used in the time embedding plot in b). b) Time embedding for the Ricker-Constant model with $\tau=6$ and $a=32.0$ (Star) or $a=33.0$ (Circle). In both plots, $\alpha=0.1$, $\beta=0.3$, $K=1.0$, and $b=200.0$
• Figure 3: a) Bifurcation diagram for the Ricker-Constant model with $\tau=3$. Vertical lines with star and circle denote the $a$ values used in the time embedding plot in b). b) Time embedding for the Ricker-Constant model with $\tau=3$ and $a=10.92$ (Star) or $a=11.04$ (Circle). Colour (yellow, green, blue, purple) refers to the ordering of $N(t)$ in the sequence. The time embedding plots were from the last 1000 time units from a 1,000,000 long simulation. In both plots, $\alpha=0.1$, $\beta=0.3$, $K=1.0$, and $b=200.0$
• Figure 4: a) Bifurcation diagram for the Ricker-Constant model with $\tau=5$. Vertical lines with star and circle denote the $a$ values for the time embedding plot in b). b) Time embedding for the Ricker-Constant model with $\tau=5$ and $a=20.2$ (Star) or $a=20.7$ (Circle). Colour (yellow, green) refers to the ordering of $N(t)$ in the sequence. The time embedding plots were from the last 1000 time units from a 1,000,000 long simulation. In both plots, $\alpha=0.1$, $\beta=0.3$, $K=1.0$, $b=200.0$
• Figure 5: a) Two parameter ($a$ & $\overline{p}$) bifurcation diagram for the Ricker-Constant model. Note that the oscillations are either the Neimark--Sacker bifurcation for even $\tau$ or the first period doubling bifurcation for odd $\tau$. The star refers to $a=13.0$ & $\overline{p}=0.57205$. The square refers to $a=13.0$ & $\overline{p}=0.35$. b) & c) Orbit diagrams with changing $\tau$ corresponding to the star and square parameter combinations respectively. In b) (star), $\overline{p}=0.57205$, and in c) (square), $\overline{p}=0.35$. Orbit diagrams were created from the last 50 time units of a 1,000,000 long simulation. For all plots, $\alpha=0.1$, $\beta=0.3$, $K=1.0$, $b=200.0$. A, B, C, D refer to the different scenarios when increasing $\tau$: (A) equilibrium changes from zero to positive, (B) dynamics become oscillatory, (C) dynamics returns equilibrium, and (D) decreasing to extinction.

Theorems & Definitions (48)

• Proposition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Theorem 3.1
• Theorem 3.2
• proof
• Lemma 3.3
• proof