A model for a population of trees structured by phenological traits

TL;DR

This work develops a population model for trees structured by two breeding-value traits, $x$ (seed production) and $y$ (summer-dormancy threshold), to study the joint effects of phenotypic plasticity and genetic evolution under climate change. Starting from a detailed annual structured integro-differential model and Fisher's Infinitesimal reproduction, the authors derive a continuous limit and, under a small-variance approximation, a macroscopic two-ODE system for the mean traits $(X(t),Y(t))$ with coefficients computed from environmental functions. Numerical simulations on fixed environments and climate-shift scenarios reveal opposing impacts of water stress and temperature stress on population dynamics, and show that plasticity in summer dormancy can reduce mortality but also slow adaptive evolution. The resulting framework provides a tractable, mechanism-based tool to predict forest adaptive capacity and to inform management strategies under changing environmental conditions.

Abstract

In the context of global warming, tree populations rely on two primary mechanisms of adaptation: phenotypic plasticity, which enables individuals to adjust their behavior in response to environmental stress, and genetic evolution, driven by natural selection and genetic diversity within the population. Understanding the interplay between these mechanisms is crucial for assessing the impacts of climate change on forest ecosystems and for informing sustainable management strategies. In this manuscript, we focus on a specific phenological adaptation: the ability of trees to enter summer dormancy once a critical temperature threshold is exceeded. Individuals are characterized by this threshold temperature and by their seed production capacity. We first establish a detailed mathematical model describing the population dynamics under these traits, and progressively reduce it to a system of two coupled ordinary differential equations. This simpler macroscopic model is then analyzed numerically, to investigate how the population reacts to a shift in its environment: an temperature increase, a drop in precipitation levels, or a combination of the two. Our results highlight contrasting effects of water stress and temperature stress on population dynamics, as well as the ambivalent effect of the plasticity.

Figures (7)

• Figure 1: Phenology of a typical tree. (a) Cases where $D(t)<W(t)$, trees produce seeds until seasonal time $s=D(t)$ when they become dormant until the next season. Note that in this case, the tree is not using all the available water. (b) Cases where $W(t)<D(t)$, the trees suffer from a lack of water from seasonal time $W(t)$ until they become dormant at seasonal time $D(t)$, this stress results in an increased mortality rate for trees during that seasonal period.
• Figure 2: Vector field for the differential equation \ref{['eq:dXY']} on $(X(t),Y(t))$. The three graphs corresponds to different values of the base death rate $d\geq 0$: (a): case where the $d=0$, (b): case where $d>0$ is positive. The blue line represents the CESS, the letters $\bf A$, $\bf B$, $\bf C$ represent the three regions described in Section \ref{['subsec:num-fixed']}, and the blue arrows represent the vector field associated to the differential equation \ref{['eq:dXY']}. Parameters:$\alpha=3, \beta=1, \gamma=2, s_0=0, \sigma_{x}=50, \sigma_y=1 \text{ and } d \in \{0,1,2\}$.
• Figure 3: Effect of an elevation of the temperatures: (a) when the base death rate is $d=0$; (b) when the base rate is positive but small $d>0$. The blue line represents the CESS before the environmental shift, while the red line and vectors represent the new CESS and the vector field of the differential equation \ref{['eq:dXY']} after the shift. The population trajectory is represented in black where the dot and the star represent initial and final positions respectively. Parameters:$\alpha=3, \beta=1, \gamma=2, s_0=0, \sigma_{x}=50, \sigma_y=1 \text{ and } d \in \{0,1\}$.
• Figure 4: Effect of a drop in precipitation levels, when only $X(t)$ evolves (that is when $\bar{\sigma}_y=0$): (a) effect of a small to moderate drop in precipitation levels (when $d=0$); (b) effect of a large drop in precipitation levels (when $d=0$). The blue line represents the CESS before the environmental shift, while the blue line and vectors represent the new CESS and the vector field of the differential equation \ref{['eq:dXY']} after the shift. The population trajectory is represented in black where the dot and star represent initial and final positions respectively. Parameters:$\alpha=3, \beta=1, \gamma=2, s_0=0, \sigma_{x}=50, \sigma_y=0$.
• Figure 5: Effect of a drop in precipitation levels, when both $X(t)$ and $Y(t)$ evolve: (a) case where $\bar{\sigma}_y>0$ is small; (b) case where $\bar{\sigma}_y>0$ is large. The blue line represents the CESS before the environmental shift, while the blue line and vectors represent the new CESS and the vector field of the differential equation \ref{['eq:dXY']} after the shift. The population trajectory is represented in black where the dot and the star represent initial and final positions respectively.. Parameters:$\alpha=3, \beta=1, \gamma=2, s_0=0, \sigma_{x}=50, \sigma_y \in \{0.1,1\} \text{ and } d=0$.

Theorems & Definitions (8)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition A.1
• Remark A.2
• proof : Proof of Proposition \ref{['prop:operatorT']}
• Lemma A.3
• proof