Ordinary Differential Equations (15): Population Dynamics

Why do lynx and snowshoe hare populations exhibit remarkable periodic fluctuations? Why does introducing a new species sometimes lead to ecological disaster? Why can some species coexist while others inevitably face competitive exclusion? The answers to these questions lie in differential equations. In this chapter, we explore the mathematical theory of population dynamics, from simple single-species growth to complex multi-species interactions, seeing how mathematics reveals the deep laws of ecosystems.

Single-Species Population Growth

Malthusian Exponential Growth Model

The simplest population model assumes constant growth rate:

whereis population size andis the intrinsic growth rate (birth rate minus death rate).

Solution:

Problem: Exponential growth is impossible in the long term — resources are limited!

Logistic Growth Model

In 1838, Belgian mathematician Pierre Verhulst proposed a more realistic model:whereis the carrying capacity, representing the maximum population the environment can support.

Key characteristics: - When: Approximately exponential growth - When: Maximum growth rate - When: Growth rate approaches zero -is a stable equilibrium

Allee Effect

In some species, too low population density actually causes decreased growth rate (difficulty finding mates, inability to defend against predators, etc.). This is called the Allee effect.

Strong Allee effect model:whereis the Allee threshold.

• If: Population declines toward extinction
• If: Population grows towardThe Allee effect explains why some endangered species are difficult to recover once their numbers become too low.

Predator-Prey Models

Lotka-Volterra Model

In 1925-1926, Lotka and Volterra independently proposed a model for predator-prey interactions: Where: -: Prey population -: Predator population -: Prey intrinsic growth rate -: Predation rate -: Predator death rate -: Energy conversion efficiency

Equilibrium analysis:

1. Trivial equilibrium: — Both extinct
2. Coexistence equilibrium:At the coexistence equilibrium, eigenvalues are: (purely imaginary)

Conclusion: The coexistence equilibrium is a center, with infinitely many closed orbits around it.

Periodic Oscillations

A striking feature of the Lotka-Volterra model is the periodic oscillation of population numbers. This explains the classic hare-lynx cycles observed in Hudson Bay Company fur records.

Functional Responses

In the classical model, predation rate is proportional to prey density. But in reality, predators have handling time and saturation effects.

Holling Type II functional response:

Holling Type III functional response:

Paradox of Enrichment

A counter-intuitive phenomenon: Increasing carrying capacitycan destabilize the system!

Asincreases: 1. Small: Stable equilibrium 2. Medium: Hopf bifurcation, stable limit cycle appears 3. Large: Limit cycle amplitude increases, potentially causing extinction

This is called the Paradox of Enrichment.

Competition Models

Lotka-Volterra Competition Equations

Two species competing for the same resource: Where: -: Competition coefficient of species 2 on species 1 -: Competition coefficient of species 1 on species 2

Four Possible Outcomes

Depending on parameter relationships, the system can have four outcomes:

1. Species 1 wins:and

2. Species 2 wins:and

3. Stable coexistence:and

4. Bistability:and Competitive exclusion principle: Two species with completely overlapping niches cannot stably coexist.

Ecological Interpretation of Coexistence

The stable coexistence conditionmeans:

Intraspecific competition is stronger than interspecific competition

Intuitively, if a species affects itself more than it affects its competitor, then when its numbers increase, it limits itself more than its opponent, leaving room for the competitor to survive.

This explains why species differentiation (occupying different niches) promotes coexistence.

Spatial Population Dynamics

Reaction-Diffusion Equations

When considering spatial distribution, population models become partial differential equations:This is the famous Fisher-KPP equation.

Traveling Wave Solutions

The Fisher-KPP equation has traveling wave solutions — populations spread through space at constant speed:Minimum wave speed.

Turing Patterns

Adding diffusion to predator-prey systems can lead to Turing instability— spatially uniform equilibria become unstable, forming spatial patterns.

Summary

In this chapter, we explored the rich mathematical world of population dynamics:

1. Single-species models: Malthus, Logistic, Allee effect
2. Predator-prey models: Lotka-Volterra, functional responses, paradox of enrichment
3. Competition models: Competitive exclusion, coexistence conditions
4. Mutualism: Stability conditions
5. Multi-species systems: Food chains, chaos
6. Spatial dynamics: Reaction-diffusion, traveling waves, Turing patterns

These models, while simplifying nature's complexity, reveal fundamental principles of ecosystems, providing theoretical foundations for conservation biology, fisheries management, and invasive species control.