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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?

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    $\begingroup$ 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... $\endgroup$ – PhysicsMathsLove Jan 25 at 18:00
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    $\begingroup$ @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. $\endgroup$ – mavavilj Jan 25 at 18:30
  • $\begingroup$ @PhysicsMathsLove: $(0, 1)$ and $(1, 0$ are not equilibria. Cheers! $\endgroup$ – Robert Lewis Jan 25 at 19:05
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    $\begingroup$ @RobertLewis why not? It causes both u_t and v_t = 0 which is the definition of a fixed point, no? $\endgroup$ – PhysicsMathsLove Jan 25 at 19:24
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    $\begingroup$ @PhysicsMathsLove: check out $u = 1$, $v = 0$; $u_t = u(v - 1) = -1 \ne 0$! $\endgroup$ – Robert Lewis Jan 25 at 19:28
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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.

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