On Lotka--Volterra Equations with Identical Minimal Intrinsic Growth Rate

Abstract

We investigate the long-run behavior of time-dependent Lotka--Volterra equations $({\rm E}_{\phi}):\frac{dx_i}{dt}=x_i(1+\phi(t)+\sum_{j=1}^na_{ij}x_j)$, $i=1,\ldots,n$, on the positive orthant. It is proved that the system $({\rm E}_{\phi})$ is decomposed into $({\rm E}_0)$ and the logistic equation $({\rm L}):\frac{dg}{dt}=g(1+\phi(t)-g)$ in the sense that $({\rm D}):\Phi(t)=g(t)\Psi(\int_0^tg(s)ds)$, where $\Phi(\cdot)$, $\Psi(\cdot)$, and $g(\cdot)$ are the solutions of $({\rm E}_{\phi})$, $({\rm E}_0)$, and (L), respectively. Suppose that $\phi(t)$ is periodic with the mean value ${\cal M}\{\phi\}>-1$. Then the existence of stable equilibrium, periodic solution, and chaotic motion for $({\rm E}_0)$ implies the existence of stable periodic solution, quasiperiodic solution, and chaotic motion for $({\rm E}_{\phi})$, respectively. The complete dynamical classification for 3-dimensional competitive system $({\rm E}_0)$ is provided in terms of competitive coefficients. There are 37 topological classes, in 34 of which any trajectory converges to an equilibrium. Among the remaining three classes, the first has a heteroclinic cycle attracting all positive points except the ray joining the origin and the positive equilibrium; the second possesses a family of periodic orbits attracting all positive points; the third class, where a family of periodic orbits, an asymptotically stable equilibrium, a heteroclinic cycle, and an invariant noncoordinate plane coexist, is a new type we find for the first time. It is shown that for 3-dimensional periodic competitive system $({\rm E}_{\phi})$, any trajectory for Poincaré mapping tends to a fixed point, a periodic orbit, an invariant minimal closed curve, or a heteroclinic cycle. All their attracting domains, which are all cone-like, are exactly described via (D) and carrying simplex theory. The dynamics can be completely classified into 37 classes corresponding to the autonomous systems. All counterparts hold when $\phi(t)$ is another minimal function, such as a quasiperiodic, almost periodic, or almost automorphic function, via (D), the classification above, and skew-product flow theory.