Population Dynamics: Lotka-Volterra & Competition
Population Dynamics: Lotka-Volterra & Competition
The Lotka-Volterra equations are the foundational predator-prey model. Prey grow exponentially without predators; predators die without prey. Their interaction creates closed orbits — sustained oscillations in which prey peaks lead predator peaks by a time lag.
The competition extension describes two species sharing a resource. Outcome — coexistence, competitive exclusion, or bistability — is determined by the relative magnitudes of inter- and intraspecific competition coefficients.
Lotka-Volterra System
$$\dot{x}=\alpha x-\beta xy,\quad \dot{y}=-\gamma y+\delta xy.$$ Interior fixed point: $x^*=\gamma/\delta$, $y^*=\alpha/\beta$.
Conserved Quantity
$$V=\delta x-\gamma\ln x+\beta y-\alpha\ln y=\text{const}$$ along every orbit. The system is conservative — no limit cycle, no spiral.
Example 1
Parameters $\alpha=1,\beta=0.1,\gamma=1.5,\delta=0.075$. Find the interior fixed point.
Solution: $x^*=\gamma/\delta=20$, $y^*=\alpha/\beta=10$.
Example 2
State the condition for stable coexistence in the two-species competition model.
Solution: Coexistence is stable when interspecific competition is weaker than intraspecific competition for both species simultaneously.
Practice
- What happens to both populations if all predators are suddenly removed?
- Sketch the phase portrait of the Lotka-Volterra system.
- State the competitive exclusion principle.
- What modifications introduce a stable limit cycle into the predator-prey model?
Show Answer Key
1. Without predators, prey grow exponentially: $\frac{dx}{dt}=\alpha x$ gives $x(t)=x_0 e^{\alpha t}$, unlimited growth until resources run out (no carrying capacity in the basic LV model).
2. Closed orbits around the coexistence equilibrium $(\delta/\beta,\,\alpha/\gamma)$. Prey axis horizontal, predator vertical. Orbits are counter-clockwise: prey peak leads predator peak by a quarter cycle.
3. Two species competing for the same niche cannot coexist indefinitely — the one with even a slight advantage will drive the other to extinction (Gause's principle). Coexistence requires niche differentiation.
4. Add a saturating functional response (Holling type II: $\frac{\beta xy}{1+\beta h x}$) or density-dependent prey growth ($\alpha x(1-x/K)$). The Rosenzweig-MacArthur model with these features can exhibit a stable limit cycle via a Hopf bifurcation.