Dynamical Systems
Population Dynamics
In 2020 SIR models and their variants became instantly famous: every conversation about "flattening the curve" and "herd immunity" was a direct application of dynamical systems. Three equations from Kermack and McKendrick in 1927 were shaping quarantine policy in 2020.
- **Fisheries:** catch quotas are calculated using logistic models - maximum sustainable yield at N = K/2
- **Epidemiology:** SIR/SEIR models were used to calculate vaccination thresholds for COVID-19 in every country
- **Conservation:** the Allee effect explains why species with small populations go extinct even when habitat is restored
Предварительные знания
Lotka-Volterra System
**Hares and lynxes in the Canadian forest show remarkably regular oscillations with a ~10-year period.** Many hares → more lynxes → fewer hares → fewer lynxes → more hares. Alfred Lotka (1910) and Vito Volterra (1926) independently proposed a mathematical model for this predator-prey dynamic.
**Lotka-Volterra system (predator-prey):** dx/dt = αx − βxy, dy/dt = δxy − γy, where x is prey and y is predator. Parameters: α - prey growth rate, β - predation rate, δ - predator efficiency, γ - predator death rate. The system is conservative: a first integral **I = δx − γ ln x + βy − α ln y = const** exists.
| Equilibrium | Coordinates | Stability |
|---|---|---|
| Trivial | (0, 0) | Unstable (both species vanish) |
| Nontrivial | (γ/δ, α/β) | Center (neutrally stable) |
| Real data | Hudson Bay hare/lynx | Oscillations ~10 years, 1845-1935 |
Volterra and the Fishing Paradox
Vito Volterra built his model in 1926 following a question from his son-in-law, marine biologist Umberto D'Ancona. D'Ancona had noticed that during World War I (when fishing was reduced) the proportion of predatory fish in catches increased. Volterra explained this mathematically: reduced fishing shifted the equilibrium toward predators. This result is known as Volterra's paradox.
In the Lotka-Volterra system the nontrivial equilibrium (x* = γ/δ, y* = α/β) is:
The Logistic Equation
**Real populations do not grow forever - resources are finite.** Malthus in 1798 proposed exponential growth dN/dt = rN. Verhulst in 1838 made the model more realistic by adding saturation: as N → K, growth slows. The logistic equation is one of the most important formulas in biology.
**Logistic equation:** dN/dt = rN(1 − N/K), where r is the intrinsic growth rate and K is the carrying capacity. Solution: **N(t) = K / (1 + ((K−N₀)/N₀)·e^{−rt})**. Two equilibria: N* = 0 (unstable) and N* = K (stable). When N₀ < K the population grows and saturates at K. S-shaped growth curve.
| Growth model | Equation | Behavior |
|---|---|---|
| Malthus (1798) | dN/dt = rN | Exponential growth, N → ∞ |
| Verhulst (1838) | dN/dt = rN(1−N/K) | S-curve, saturation at K |
| Gompertz | dN/dt = rN·ln(K/N) | Asymmetric S-curve |
| Allee effect | dN/dt = rN(N−A)(1−N/K) | Critical density A |
**The Allee effect**: a modification of the logistic model: at small N < A the population goes extinct because individuals cannot find mates. This creates a **threshold effect**: populations below the critical level A irreversibly go extinct even when resources K >> A are available. Important for species conservation.
In the logistic equation, the growth rate dN/dt is maximum at N₀ = K/2. Why?
SIR Epidemic Model
**Why do some epidemics spread while others fade out on their own?** The SIR model - three equations that predict the fate of an outbreak better than intuition. The key parameter is the basic reproduction number R₀: if R₀ > 1, the epidemic spreads; if R₀ < 1, it fades.
**SIR model** (Kermack-McKendrick, 1927): dS/dt = −βSI/N, dI/dt = βSI/N − γI, dR/dt = γI. S - susceptible, I - infected, R - recovered. **R₀ = β/γ**: basic reproduction number. Threshold theorem: the epidemic spreads if and only if **S₀ > γ/β = N/R₀**. Herd immunity is reached at immune fraction **p_c = 1 − 1/R₀**.
| Disease | R₀ | Herd immunity threshold |
|---|---|---|
| Seasonal flu | 1.2 - 1.4 | 17% - 29% |
| COVID-19 (original) | 2.0 - 3.0 | 50% - 67% |
| Smallpox | 5 - 7 | 80% - 85% |
| Measles | 12 - 18 | 92% - 94% |
For measles R₀ ≈ 15. What fraction of the population must be immune to achieve herd immunity?
Realistic Predator-Prey Models
**The simple Lotka-Volterra system is just a starting point.** Real ecosystems are more complex: predator saturation (Holling functional response), logistic prey growth, interspecific competition. Adding realism transforms the conservative system into a dissipative one - with stable limit cycles and potential chaos.
**Rosenzweig-MacArthur model**: a realistic predator-prey system: **dx/dt = rx(1−x/K) − axy/(1+ahx), dy/dt = eaxy/(1+ahx) − dy**. Type II functional response: ax/(1+ahx) - predator saturation. Parameters: r - prey growth, K - carrying capacity, a - search rate, h - handling time, e - efficiency, d - predator death rate.
| Model type | Features | Dynamics |
|---|---|---|
| Classic Lotka-Volterra | Linear prey growth, linear predation | Conservative oscillations |
| With logistic prey growth | Prey saturates at K | Spiral convergence to equilibrium |
| With Holling II response | Predator saturates | Limit cycle or equilibrium |
| With Holling III response | Sigmoidal saturation | Multiple equilibria possible |
The Paradox of Enrichment
In 1971 Martin Rosenzweig discovered the "paradox of enrichment": increasing the carrying capacity K (e.g. through fertilization) does not improve ecosystem stability - it worsens it. Above a critical K a Hopf bifurcation occurs and the system transitions to large-amplitude oscillations that can drive both species to extinction. This turned the traditional ecological view of "enrichment" upside down.
More resources = more stable ecosystem
In models with predator saturation (Holling II), increasing carrying capacity K destabilizes the system through a Hopf bifurcation. This is the paradox of enrichment.
At large K the predator rarely experiences prey shortage → predator population grows → prey crashes sharply → predator crashes sharply → cycle repeats with growing amplitude. Nonlinear feedback through saturation creates instability where linear logic would predict stability.
In the Rosenzweig-MacArthur model, increasing the carrying capacity K beyond the critical value causes:
Key Takeaways
- **Lotka-Volterra system** is conservative: first integral I = const; oscillations are eternal and undamped; nontrivial equilibrium is a center
- **Logistic equation** dN/dt = rN(1−N/K): S-shaped growth curve, saturation at carrying capacity K, maximum growth at N = K/2
- **SIR model:** R₀ = β/γ determines the fate of the epidemic; herd immunity at fraction p_c = 1 − 1/R₀
- **Paradox of enrichment:** increasing K in realistic models destabilizes the system through a Hopf bifurcation
Related Topics
Population dynamics is a living example of all nonlinear dynamics concepts:
- Bifurcations — The paradox of enrichment is a Hopf bifurcation; the epidemic threshold R₀ = 1 is a transcritical bifurcation
- Neurodynamics — Neural population equations are analogous to Lotka-Volterra - the Wilson-Cowan equations have the same structure
- Climate Models — Climate tipping points are analogues of herd immunity thresholds and the paradox of enrichment
Вопросы для размышления
- Classic Lotka-Volterra predicts eternal, undamped oscillations. Real data shows damped oscillations or limit cycles. What additions to the model make it realistic?
- R₀ depends on human behavior, not just biology. How does changing behavior (masks, distancing) affect R₀ in the SIR model?
- The Allee effect creates threshold instability - populations below the threshold go extinct. Which species are most vulnerable to this, and why?