# Lotka-Volterra: is stability analysis done on both equations separately or to their sum?

Lotka-Volterra: is stability analysis done on both equations separately or to their sum?

So if the systems are e.g. notated as:

$$u_t=u(v-1)$$ $$v_t=v(1-u)$$

then would one do stability analysis for $$u_t$$ and then for $$v_t$$ or would one consider

$$u_t+v_t=0$$

or

$$u_t=v_t=0 \implies u_t-v_t=0$$

and then do the analysis?

• I would do it for both at the same time. The fixed points satisfy $u(v-1) = 0$ and $v(1-u) = 0$ so either $u=0$ or $v=1$ or $v= 0$ or $u = 1$, giving 4 points $(0, 0)$, $(0, 1)$, $(1, 0)$ and $(1, 1)$. Then you can calculate the Jacobean and evaluate the stability. However, I am a student so there may be other ways... – PhysicsMathsLove Jan 25 at 18:00
• @PhysicsMathsLove This www2.hawaii.edu/~taylor/z652/PredatorPreyModels.pdf has an example where they calculate the stationary points so that each equation produces the other point. Then they form the Jacobian based on these points. – mavavilj Jan 25 at 18:30
• @PhysicsMathsLove: $(0, 1)$ and $(1, 0$ are not equilibria. Cheers! – Robert Lewis Jan 25 at 19:05
• @RobertLewis why not? It causes both u_t and v_t = 0 which is the definition of a fixed point, no? – PhysicsMathsLove Jan 25 at 19:24
• @PhysicsMathsLove: check out $u = 1$, $v = 0$; $u_t = u(v - 1) = -1 \ne 0$! – Robert Lewis Jan 25 at 19:28

Neither considering the variables separately or analyzing the sum variable $$u_t + v_t$$ will resolve the stability issues for the system

$$u_t = u(v - 1), \tag 1$$

$$v_t = v(1 - u). \tag 2$$

Considering the variables separately fails for the reason that $$u$$ and $$v$$ are coupled in the system (1)-(2); the time evolution of $$u(t)$$ affects the time evolution of $$v(t)$$ and vice-versa; we can't determine one without the other. And while it is legitamite to form the sum $$u_t + v_t$$, in the absence of a second variable such as perhaps $$u(t)$$, $$v(t)$$, or $$u_t - v_t$$, the system is incomplete and will lack sufficient information to determine its evolution. Furthermore, it is not clear that any useful simplification will be had by re-writing (1)-(2) in terms of the variable $$u_t + v_t$$; one would have to present a specific equation for $$u_t + v_t$$ to resolve such a question.

One needs to consider (1)-(2) as a single entity in the vector variable

$$\mathbf r(u, v) = \begin{pmatrix} u \\ v \end{pmatrix}; \tag 3$$

in so doing, we may present the right-hand sides of (1)-(2) as the vector field

$$\mathbf X(u, v) = \begin{pmatrix} u(v - 1) \\ v(1 - u) \end{pmatrix}; \tag 4$$

the given system is then written in vector form

$$\dot{\mathbf r}(u, v) = \mathbf X(u, v). \tag 5$$

We may find the equilibria of (1)-(2), (5) by solving

$$u_t = u(v - 1) = 0, \tag 6$$

$$v_t = v(1 - u) = 0 \tag 7$$

simultaneously for $$u$$, $$v$$; we see that if either

$$u = 0 \; \text{or} \; v = 0 \tag 8$$

then

$$u = v = 0; \tag 9$$

also, from (6) and (7),

$$u \ne 0 \Longrightarrow v = 1 \Longrightarrow u = 1; \tag{10}$$

thus another equilibrium point is

$$u = v = 1; \tag{11}$$

it is easy to see there are no others.

We next linearize the system about the equilibria $$(0, 0)$$, $$(1, 1)$$; the Jacobean matrix of first derivatives is

$$J_{\mathbf X}(u, v) = \begin{bmatrix} u_{t, u} & u_{t, v} \\ v_{t, u} & v_{t, v} \end{bmatrix}, \tag{12}$$

where I have introduced the shorthand notation

$$u_{t, u} = \dfrac{\partial u_t}{\partial u}, \tag{13}$$

and so forth; then

$$J_{\mathbf X}(u, v) = \begin{bmatrix} v - 1 & u \\ -v & 1 - u \end{bmatrix}; \tag{14}$$

at $$(0, 0)$$,

$$J_{\mathbf X}(0, 0) = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}; \tag{15}$$

the eigenvalues of this matrix are clearly $$\pm 1$$; $$(0, 0)$$ is thus a saddle point, an equilibrium with both stable an unstable trajectories nearby, hence unstable as a critical point of (1)-(2); as for $$(1, 1)$$ we have

$$J_{\mathbf X}(1, 1) = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}, \tag{16}$$

the eigenvalues of which are easily seen to be $$\pm i$$; we have in $$(1, 1)$$ a center, indicating the possible presence of periodic orbits surrounding $$(1, 1)$$, though the eigenvalues alone are not decisive in this case.

The preceding calculations show what is done in a typical, first order, elementary stability analysis for the system (1)-(2), (5). At a more advanced level, we might address the stability of non-equilibrium orbits, that is, whether trajectories near a given one $$(u(t), v(t))$$ converge to, or diverge from, it. But such an undertaking, as well as the determination of the existence of truly periodic orbits, is a more advanced and involved undertaking which will be reserved for perhaps future posts on the subject.