Solving the nonlinear differential system $x'(t)=x(t)(s(t)-a)$, $y'(t)=y(t)(s(t)-b)$, $z'(t)=z(t)(s(t)-c)$, where $s(t)=ax(t)+by(t)+cz(t)$

Question

Is it possible to solve the following system of differential equations? If so, how might the solution be approached? I am not sure what class of functions these fall into (if any), and I am not familiar with solving systems of ODE's so all help is appreciated.

$$ x'(t) = -ax(t)+ax(t)x(t)+by(t)x(t)+cz(t)x(t) \\ y'(t) = -by(t)+ax(t)y(t)+by(t)y(t)+cz(t)y(t) \\ z'(t) = -cz(t)+ax(t)z(t)+by(t)z(t)+cz(t)z(t) \\ \text{ } \\ x(t) + y(t) + z(t) = 1, \ \ \ \ \forall \ t \\ \text{ }\\ \text{where $a,b,c$ are constants, and $x,y,z$ are functions of $t$} \\ \text{and initial values $x(0)$, $y(0)$, $z(0)$ are known} \\ \text{ } $$

Answer 1

Since $z(t)=1-x(t)-y(t)$ for every $t$, this is equivalent to the 2D system $$x'=ux(1-x)-vxy\qquad y'=vy(1-y)-uxy$$ with $$u=c-a\qquad v=c-b$$ From this point on, the behaviour is a matter of signs, of the parameters $(u,v)$ and of the initial conditions $(x(0),y(0))$.

One can suspect that you have in mind the case where $x(0)$, $y(0)$, $z(0)$ are the proportions of a tripartite population, hence nonnegative (and summing to $1$). Then, if $u>0$ and $v>0$, one is modelling a competition between some species $x$ and $y$, each with a logistic dynamics when in isolation, and the usual tools (mainly a planar phase diagram, including the isoclines and the fixed points, with their types) allow to compute its asymptotics.

Still if $u>0$ and $v>0$ (that is, $c>b$ and $c>a$), the species with the higher Malthusian parameter survives, that is:

• If $u>v$ ($a<b$) then $(x(t),y(t))\to(1,0)$
• If $u<v$ ($a>b$) then $(x(t),y(t))\to(0,1)$
• If $u=v$ ($a=b$), then $(x(t),y(t))\to(x_\infty,y_\infty)$ where $x_\infty+y_\infty=1$ and $(x_\infty,y_\infty)$ is proportional to $(x(0),y(0))$, thus, $x_\infty=\frac{x(0)}{1-z(0)}$ and $y_\infty=\frac{y(0)}{1-z(0)}$.

One can check the so-called competitive exclusion principle, which stipulates that, in generic competition systems, exactly one species survives. Here, when $(a,b,c)$ are positive and distinct, the species with the smallest $(a,b,c)$ parameter survives (proportion going to $1$) and the two others die (proportions both going to $0$).

Phase diagram for $(u,v)=(2,3)$ (every $(u,v)$ with the same ratio $u/v$ yields the same diagram, only the speed of the dynamics changes, the case $u>v$ being symmetrical):

Phase diagram for $(u,v)=(3,3)$ (the diagrams for every $u=v>0$ coincide, only the speed of the dynamics changes):

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Edit: For a more direct, general, and powerful approach, note that this is a special case of an $n$-dimensional system $$x'_i(t)=-a_ix_i(t)+s(t)x_i(t)\qquad s(t)=\sum\limits_{i=1}^na_ix_i(t)$$ If $\sum\limits_{i=1}^nx_i(0)=1$, then $\sum\limits_{i=1}^nx_i(t)=1$ for every $t$, and $$x_i(t)=x_i(0)e^{-a_it}e^{S(t)}\qquad S(t)=\int_0^ts(\tau)d\tau$$ hence $$x_i(t)=\frac{x_i(0)e^{-a_it}}{\sum\limits_{k=1}^nx_k(0)e^{-a_kt}}$$ for every $i$, and the results above follow in this more general setting, for example the fact that if $a_i<a_j$ for some $i$ and every $j\ne i$, then the species $i$ becomes dominant in the sense that $x_i(t)\to1$ and $x_j(t)\to0$ for every $j\ne i$.