Applied Mathematics and Numerical Analysis Seminar

25/01/2023, 16:00 — 17:00 — Room P3.10, Mathematics Building Victor Ortega, Departmento de Matemática Aplicada, Universidad de Granada, Spain and CEMAT, Faculdade de Ciências, Universidade de Lisboa, Portugal

Some stability criteria in the periodic prey-predator Lotka-Volterra model

In this talk, we present some stability results in a classical model concerning population dynamics, the nonautonomous prey-predator Lotka-Volterra model under the assumption that the coefficients are $T$-periodic functions \label{sysLV}
\left\lbrace \begin{array}{l} \dot{u}=
u(a(t) - b(t)\,u - c(t)\,v), \\ \dot{v}=
v(d(t) + e(t)\,u - f(t)\,v), \end{array} \right.
where $u\gt 0$, $v\gt 0$. The variables $u$ and $v$ represent the population of a prey and its predator, respectively. Some instances with this kind of dynamics can be: snowshoe hare and lynx canadensis, paramecium and didinium, fish population and fishermen, etc. The periodicity of this model takes into account changes of the environment in which the predation process takes place. For instance, seasonality or variations of the temperature in laboratory conditions.

In the system \eqref{sysLV} the coefficients $b(t)$, $c(t)$, $e(t)$ and $f(t)$ are positive. The coefficients $c(t)$ and $e(t)$ describe the interaction between $u$ and $v$; $a(t)$ and $b(t)$ describe the growth rate for the prey $u$; $d(t)$ and $f(t)$ represent the analogous for the predator $v$.

Solutions for the system \eqref{sysLV} with both components positive are called coexistence states and the necessary and sufficient conditions for their existence are well understood, see [2].

After reviewing those conditions, we present some results concerning the stability of a special kind of coexistence state, positive $T$-periodic solutions. In [3] the author gave a sufficient condition for the uniqueness and asymptotic stability of the positive $T$-periodic solution. This criterion is formulated in terms of the $L^1$ norm of the coefficients of a planar linear system associated to \eqref{sysLV}. On the other hand, in [1], assuming that the system \eqref{sysLV} has no sub-harmonic solutions of second order (periodic solutions with minimal period $2T$), the authors proved that there exists at least one asymptotically stable $T$-periodic solution. Here the result is formulated in terms of the $L^\infty$ norm. Our result, in [4], gives a $L^p$ criterion, building a bridge between the two previous results.

This is a Joint work with Carlota Rebelo (Departmento de Matemática and CEMAT, Faculdade de Ciências, Universidade de Lisboa, Portugal).

Acknowledgements: This work was partially supported by the Spanish Ministerio de Universidades and Next Generation Funds of the European Union.

References

1. Z. Amine, R. Ortega, A periodic prey-predator system, Journal of Mathematical Analysis and Applications,185(2): 477-489, 1994.
2. J. López-Gómez, R. Ortega and A. Tineo, The periodic predator-prey Lotka-Volterra model, Adv. Differential Equations, 1(3): 403-423, 1996.
3. R. Ortega, Variations on Lyapunov's stability criterion and periodic prey-predator systems, Electronic Research Archive, 29(6): 3995-4008, 2021.
4. V. Ortega, C. Rebelo, A $L^p$ stability criterion in the periodic prey-predator Lotka-Volterra model, In preparation, 2023.