Stoichiometric ontogenetic development influences population dynamics: Stage-structured model under nutrient co-limitations

TL;DR

The paper tackles how nutrient quantity and quality interact with stage-structured life history to shape population dynamics in a nutrient-co-limited ecosystem. It develops a co-limited, stage-structured producer–grazer model (algae–Daphnia) that embeds P:C–based food quality into maturation and growth through smooth co-limitation functions, with Q = (P - theta_j J - theta_a A)/x and h_i(Q) = tanh(Q/theta_i). The authors prove positivity and boundedness, analyze grazer-extinction equilibria E1 = (K,0,0) and E2 = (P/q,0,0), and perform numerical bifurcation and sensitivity analyses, revealing Hopf and saddle-node transitions and the strong influence of both environmental parameters (P, K) and stage-specific traits (c_j, c_a, e_j, e_a, theta_j, theta_a, delta_j, delta_a) on dynamics. The findings show that stage-structured stoichiometric constraints can shift equilibria and generate cycles, with effects amplified under high-light conditions when food quality declines, highlighting key mechanisms by which ontogeny and nutrient co-limitation govern population outcomes.

Abstract

Ecological processes depend on the flow and balance of essential elements such as carbon (C) and phosphorus (P), and changes in these elements can cause adverse effects to ecosystems. The theory of Ecological Stoichiometry offers a conceptual framework to investigate the impact of elemental imbalances on structured populations while simultaneously considering how ecological structures regulate nutrient cycling and ecosystem processes. While there have been significant advances in the development of stoichiometric food web models, these efforts often consider a homogeneous population and neglect stage-structure. The development of stage-structured population models has significantly contributed to understanding energy flow and population dynamics of ecological systems. However, stage structure models fail to consider food quality in addition to food quantity. We develop a stoichiometric stage-structure producer-grazer model that considers co-limitation of nutrients, and parameterize the model for an algae-Daphnia food chain. Our findings emphasize the impact of stoichiometric constraints on structured population dynamics. By incorporating both food quantity and quality into maturation rates, we demonstrate how stage-structured dynamics can influence outcomes in variable environments. Stage-specific parameters, such as juvenile growth and ingestion rates can drive shifts in equilibria, limit cycles, and bifurcation points. These effects are especially significant in high-light environments where nutrient limitations are most pronounced.

Figures (10)

• Figure 1: Visualizations of nonsmooth functional forms using Leibig minimum approach vs smooth co-limitation approach: (a) producer growth with minimum function $\min\left\{1-\frac{x}{K},1-\frac{q}{Q}\right\}$, (b) producer growth using multiplicative co-limitation function $\left(1-\frac{x}{K}\right)\left(1-\frac{q}{Q}\right)$, (c) grazer conversion efficiencies. Here we use $K=2$ mg C/l, $q=0.0038$ mg P/mg C, $\theta=0.03$ mg P/mg C.
• Figure 2: Numerical simulation of the model \ref{['eq:Model']} showing population densities (black) and the proportion of C in the grazer population composed of adults (red), for parameter values listed Table \ref{['tab:params']} and varying values for $K$ representing (a) low light levels $K=0.5$ mg C/l, (b) medium light levels $K=1$ mg C/l, and (c) high light levels $K=2$ mg C/l. Initial conditions are $x(0)=0.5$ mg C/l, $J(0)= 0.125$ mg C/l, and $A(0)=0.125$ mg C/l.
• Figure 3: Numerical simulation of the model \ref{['eq:Model']} showing $Q$, the variable P:C ratio of the producer, for parameter values listed Table \ref{['tab:params']} and varying values for $K$. Initial conditions are $x(0)=0.5$ mg C/l, $J(0)= 0.125$ mg C/l, and $A(0)=0.125$ mg C/l. Simulations correspond with those in Fig. \ref{['fig:sims']}.
• Figure 4: Bifurcation diagram of the model \ref{['eq:Model']} using parameters in Table \ref{['tab:params']} and bifurcation parameter $K$. These bifurcation diagrams show the long-term behavior ($t>1500$ days) of the juvenile density (left column), the adult density (middle column), and total grazer density (right column) for varying values of the juvenile P:C ratio $\theta_j$ (top row), juvenile maximum conversion efficiency $e_j$ (middle row), and juvenile maximum ingestion rate $c_j$ (bottom row).
• Figure 5: Two parameter bifurcation showcasing long term dynamics ( $>5000$ days) of (a) Juvenile and (b) Adult population densities when light-dependent producer carrying capacity $K=0.5$ mg C/l; and (c) Juvenile and (d) Adult population densities when light-dependent producer carrying capacity $K=1$ mg C/l for varying values of juvenile maximum ingestion rate $c_j$ and juvenile maximum conversion efficiency $e_j$. Black curves separates regions where long term dynamics approach equilibria and limit cycles. The black curve in (a) and (b) corresponds with the Hopf bifurcation. The black curve in (c) and (d) corresponds with the collapse of high amplitude cycles at the saddle-node bifurcation. Region 1 exhibits equilibria dynamics and Region 2 exhibits limit cycles, where the average density of the limit cycles is shown in the heatmap.

Theorems & Definitions (4)

• Theorem 3.1
• proof
• Theorem 3.2
• proof