Lotka–Volterra equations
The previous lesson introduced the biological predator–prey feedback. Here the emphasis is different: we analyse the classical Lotka–Volterra system as a nonlinear dynamical system.
The classical system
\[\boxed{\begin{aligned}\frac{dN}{dt}&=\alpha N-\beta NP,\\[3pt]\frac{dP}{dt}&=\delta NP-\gamma P.\end{aligned}}\]Here \(N\) is prey abundance, \(P\) is predator abundance, and all four parameters are positive.
Factoring gives
\[\frac{dN}{dt}=N(\alpha-\beta P),\qquad \frac{dP}{dt}=P(\delta N-\gamma).\]Nullclines
The prey nullclines satisfy \(dN/dt=0\):
\[N=0\qquad\text{or}\qquad P=\frac{\alpha}{\beta}.\]The predator nullclines satisfy \(dP/dt=0\):
\[P=0\qquad\text{or}\qquad N=\frac{\gamma}{\delta}.\]Equilibria
The two equilibria are
\[(0,0)\]and
\[\boxed{\left(N^*,P^*\right)=\left(\frac{\gamma}{\delta},\frac{\alpha}{\beta}\right)}.\]The second equilibrium represents positive coexistence.
Direction of motion
| Region | Prey | Predators |
|---|---|---|
| \(P<\alpha/\beta\), \(N>\gamma/\delta\) | increase | increase |
| \(P>\alpha/\beta\), \(N>\gamma/\delta\) | decrease | increase |
| \(P>\alpha/\beta\), \(N<\gamma/\delta\) | decrease | decrease |
| \(P<\alpha/\beta\), \(N<\gamma/\delta\) | increase | decrease |
These sign changes produce rotation around the positive equilibrium.
The Jacobian matrix
For
\[f(N,P)=\alpha N-\beta NP,\qquad g(N,P)=\delta NP-\gamma P,\]the Jacobian is
\[\boxed{J(N,P)=\begin{pmatrix}\alpha-\beta P&-\beta N\\\delta P&\delta N-\gamma\end{pmatrix}}.\]It describes the first-order behaviour of the system near an equilibrium.
Linearisation at coexistence
At \(N^*=\gamma/\delta\) and \(P^*=\alpha/\beta\),
\[J(N^*,P^*)=\begin{pmatrix}0&-\dfrac{\beta\gamma}{\delta}\\[6pt]\dfrac{\alpha\delta}{\beta}&0\end{pmatrix}.\]The characteristic equation is
\[\lambda^2+\alpha\gamma=0,\]so
\[\boxed{\lambda=\pm i\sqrt{\alpha\gamma}}.\]Neutral stability
For the classical Lotka–Volterra system, the positive equilibrium is a nonlinear centre. Positive trajectories around it are closed orbits rather than spirals.
This is called neutral stability: a perturbation generally moves the system onto a different closed orbit instead of making it return to the original equilibrium.
A conserved quantity
For positive \(N\) and \(P\),
\[\boxed{H(N,P)=\delta N-\gamma\ln N+\beta P-\alpha\ln P}\]is constant along a trajectory.
Differentiating along a solution gives
\[\frac{dH}{dt}=\left(\delta-\frac{\gamma}{N}\right)\frac{dN}{dt}+\left(\beta-\frac{\alpha}{P}\right)\frac{dP}{dt}.\]Substituting the differential equations gives
\[\boxed{\frac{dH}{dt}=0}.\]Each trajectory therefore remains on one level set \(H(N,P)=C\).
Why the orbits are closed
In the positive quadrant, the relevant level sets of the conserved quantity surround the coexistence equilibrium. A solution stays on its own level set, producing a closed orbit.
Different initial conditions usually give different values of \(C\), so the classical model has a family of cycles rather than one attracting cycle.
What the classical system does not predict
The model does not predict damping toward coexistence, and it has no attracting limit cycle. Those behaviours require additional mechanisms such as prey self-limitation, saturating predation, delays or other biological effects.
Connection to the next lesson
The interaction term \(\beta NP\) assumes that prey removal per predator increases linearly with prey abundance. Real predators have finite handling time and can saturate.
The next lesson develops functional responses, replacing this simple interaction assumption with more realistic consumption functions.