Mathematical Modeling in the 2024 MCM

In the 2024 Mathematical Contest in Modeling (MCM), we explored the dynamics of ecosystems with a focus on the sea lamprey (Petromyzon marinus). Our study aimed to investigate the relationship between local resource availability, sex ratio variations, and their ecological effects. Here, I will introduce the mathematical models we used and some of their key equations.

Aquatic Nutritional Network (ANN) Model

The Aquatic Nutritional Network (ANN) model helps us understand how food resources influence the sex ratio in fish populations. This model includes factors like larval growth, algal reproduction, and the dynamics of small and large fish populations.

Key Equations in the ANN Model

Larvae Population Dynamics: \(\frac{dL}{dt} = \alpha_{m}\alpha_{f}N_{m}(t)N_{p}(t) + a_{l}A(t) - mL(t)\) This equation describes the change in larvae population over time, considering growth contributions from male nutrients and algae consumption.

To solve for the equilibrium state where the change in larvae population over time is zero: \(\frac{dL}{dt} = 0 \implies \alpha_{m}\alpha_{f}N_{m}(t)N_{p}(t) + a_{l}A(t) - mL(t) = 0\) Solving for \(L(t)\): \(L(t) = \frac{\alpha_{m}\alpha_{f}N_{m}(t)N_{p}(t) + a_{l}A(t)}{m}\)

Algae Population Dynamics: \(\frac{dA}{dt} = c_{l} - a_{l}A(t)\) This equation models the growth rate of algae and its consumption by larvae.

To solve for the equilibrium state where the change in algae population over time is zero: \(\frac{dA}{dt} = 0 \implies c_{l} - a_{l}A(t) = 0\) Solving for (A(t)): \(A(t) = \frac{c_{l}}{a_{l}}\)

Small Fish Population Dynamics: \(\frac{dF_s}{dt} = c_{s} - (a_{m} + a_{f})F_s(t) + e_{s}F_s(t)\) This equation accounts for the intrinsic growth rate of small fish, parasitism by larvae, and the environmental loss factor affecting small fish.

To solve for the equilibrium state where the change in small fish population over time is zero: \(\frac{dF_s}{dt} = 0 \implies c_{s} - (a_{m} + a_{f})F_s(t) + e_{s}F_s(t) = 0\) Solving for \(F_s(t)\): \(F_s(t) = \frac{c_{s}}{(a_{m} + a_{f}) - e_{s}}\)

By solving these equations, we can better understand the equilibrium states of various populations within the ecosystem. These insights are crucial for predicting how changes in environmental factors or population dynamics can affect the overall stability and health of the ecosystem.

Determining Equilibrium Points for the ANN Model

To find the equilibrium points of the ANN model, we follow a systematic approach. At equilibrium, the rate of change for each population is zero, meaning the populations remain constant over time.

Setting Up the Equilibrium System

We transform the system’s differential equations into algebraic equations to find the equilibrium points. Each equation represents a balance within a specific ecological niche:

\[\begin{align*} \left\{ \begin{array}{l} \alpha_{m}\alpha_{f}N_{m}(t)N_{p}(t) + a_{l}A(t) - mL(t) = 0, \\ c_{l} - a_{l}A(t) = 0, \\ c_{s} - (a_{m} + a_{f})F_s(t) + e_{s}F_s(t) = 0, \\ b_{m}N_{m}(t) + b_{f}N_{f}(t) - c_{f}F_b(t) + e_{b}F_b(t) = 0, \\ a_{m}F_s(t) + P(t)L(t) - b_{m}N_m(t) = 0, \\ a_{f}F_s(t) + (1 - P(t))L(t) - b_{f}N_f(t) = 0. \end{array} \right. \end{align*}\]

Application of Constraints

Biological systems are often constrained by ecological realities. Population sizes are generally within certain bounds, such as carrying capacities or resource limitations:

\[0 \leq N_m, N_p, A, L, F_s, F_b \leq 1\]

All variables are normalized to the interval [0,1], representing proportions of maximum possible population sizes.

Numerical Solution of the Equilibrium System

By determining these equilibrium points, we gain critical insights into the sustainability and resilience of the ecosystem, with implications for species conservation and resource management strategies.

After applying this models, we can better understand the ecological impacts of sex ratio variations and resource availability, which is crucial for conservation efforts and maintaining ecological balance.