Differential Equation Modeling of Inflows/Outflows: Lamprey Ecosystem

Differential equation modeling is widely used across scientific fields. Its core idea is to describe system dynamics by modeling the rates of change of variables, allowing analysis of system behavior.

Core Idea of Differential Equation Modeling

The core of differential equation modeling lies in using differential equations to describe the change rates of variables in a system. Solving these equations enables prediction of future system behavior.

Change rate refers to the amount a variable changes per unit time, typically expressed by a derivative. For example, if we denote a quantity at time t as x(t) , its change rate is dx/dt .

This method is especially suitable for describing continuous processes in physics, chemistry, biology, and other domains. Discrete changes can be modeled with difference equations or recurrence relations, following a similar principle.

In many real problems, system variables are influenced by both external inputs ( inflow ) and outputs ( outflow ). Inflow increases the quantity of a variable, while outflow decreases it. Accurately capturing inflow and outflow simplifies model construction.

Figure 1: Reservoir analogy

Simply put, the system can be visualized as a water tank where:

Inflow is the water entering the tank (e.g., rain or river inflow), increasing the water volume.

Outflow is the water leaving the tank (e.g., evaporation, seepage, drainage), decreasing the volume.

Mathematically, if V(t) denotes the water volume at time t , with inflow rate I(t) and outflow rate O(t) , the change rate of volume is dV/dt = I(t) - O(t) .

Similarly, in ecological systems we can treat resources or population numbers as the tank volume, using inflow and outflow to describe dynamics.

Case Study: Sex Ratio Changes in Lamprey Ecosystem

The problem originates from the 2024 MCM Problem A, with modeling methods referencing award-winning paper 2405424.

The lamprey is an ancient species that can adjust its sex ratio based on resource availability.

Figure 2: Lamprey morphology

This sex‑ratio adjustment significantly impacts ecosystem stability. By building a differential equation model, we can analyze how lamprey sex‑ratio changes affect the ecosystem.

Basic Information about Lamprey

Key characteristics:

Body: elongated, eel‑like, soft, scaleless.

Mouth: suction cup with keratinous teeth for attaching to hosts and feeding on blood.

Gill openings: seven pairs on each side of the body.

Skeleton: primarily cartilage, lacking true jaws.

Figure 3: Lamprey

The lamprey life cycle includes freshwater and marine stages.

Freshwater stage (spawning and larvae):

Spawning: adults lay eggs in riverbeds, digging shallow pits.

Hatching: eggs develop into larvae (ammocoetes) that filter organic particles in sediment for 3–7 years.

Marine stage (adult):

Metamorphosis: larvae transform into parasitic adults.

Parasitism: adults attach to other fish with their suction cup mouth, feeding on blood and body fluids for 1–2 years.

Return to spawn: after a marine period, adults migrate back to freshwater to reproduce.

Lampreys can dynamically adjust their sex ratio: abundant resources favor a higher proportion of females, while scarce resources favor more males, aiding population survival under varying conditions.

Resource abundant: female proportion increases.

Resource scarce: male proportion increases.

Model Construction

The core idea of differential equation modeling uses “inflow” to represent factors that increase a variable and “outflow” for factors that decrease it. Applying this to the lamprey ecosystem yields clearer models.

Based on the Lotka‑Volterra predator‑prey model and the Nicholson‑Bailey host‑parasite model, paper 2405424 proposes three models: an ideal reference model, a constant‑sex‑ratio model, and a variable‑sex‑ratio model. Here we discuss the ideal reference model and the variable‑sex‑ratio model.

Ideal Reference Model

The ideal reference model assumes no lampreys, only resources and other species competing. Resource and species dynamics are described by inflow and outflow processes.

Figure 4: Species‑environment interaction (reference)

Variables and parameters:

R(t) : resource amount at time t .

N_i(t) : population of species i at time t .

a : natural growth rate of resources (inflow).

b : internal competition of resources (outflow).

c_i : consumption rate of species i on resources (outflow).

d_i : natural decay rate of species i (outflow).

e_i : resource utilization efficiency of species i (inflow).

f_i : intraspecific competition of species i (outflow).

The resource change rate is determined by natural growth (inflow), internal competition (outflow), and consumption by species (outflow). The species population change rate is determined by natural decay (outflow), resource utilization (inflow), and intraspecific competition (outflow).

Variable Sex‑Ratio Model

This model assumes the lamprey sex ratio dynamically adjusts with resource availability: abundant resources increase female proportion, scarce resources increase male proportion.

Figure 5: Factors affecting lamprey numbers (reference)

Variables and parameters:

R̄(t) : average resource holding per lamprey at time t .

R_min : minimum resource required for lamprey survival.

k : logistic curve stretch factor for the sex‑ratio function.

S(R) : male proportion as a function of resource level R .

Other variables follow those in the ideal reference model.

The relationship between sex ratio S and average resource holding R̄ follows a logistic curve. The average resource holding can be expressed accordingly, and the lamprey population change is driven by sex ratio and natural growth (inflow).

Integrating the variable sex‑ratio into the ideal reference model yields a system of differential equations that capture the coupled dynamics of resources, other species, and lamprey population.

Simulation results show:

The lamprey acts as a key ecological factor; its sex‑ratio regulation directly influences resource consumption and the survival of other species.

With a constant sex ratio, lamprey numbers remain stable but increase competition pressure on other species, reducing their populations and intensifying resource depletion.

A dynamically varying sex ratio adds system complexity, allowing lamprey populations to better adapt to environmental changes, though it also raises overall system volatility.

Through the inflow‑outflow perspective, the differential equation models for the lamprey ecosystem illustrate a modeling approach that can be extended to other species and ecological systems.

By employing differential equation modeling and the inflow‑outflow analysis, we gain clearer directions for constructing such models.

References:

【1】 MathModels.org. "MCM Problem A: Resource Availability and Sex Ratios." 2024. https://www.mathmodels.org/Problems/2024/MCM-A/index.html

【2】 MathModels.org. "2405424.pdf." 2024. http://www.mathmodels.org/mathmodels/2024/2405424.pdf