# Numerical integration of structured problems. The dynamics of a consumer–resource model from an evolutionary point of view

We have studied the numerical integration of a nonlinear model that describes the dynamics of size-structured populations feeding on an unstructured dynamical food source, given by the following equations,

\[ \begin{gather} \begin{aligned} & u_t + (g(x, t, s(t)) u)_x = −\mu(x, t, s(t)) u, \quad 0 < x < x_M (t), \; t > 0, \\ & g(0, t, s(t)) u(0, t) = \int_0^{x_M(t)} \alpha(x, t, s(t)) u(x, t) dx, \quad t > 0, \\ & u(x, 0) = u_0 (x), \quad 0 \leq x \leq x_M (0), \end{aligned} \\[1em] \begin{aligned} & s(t) = f (t, s(t), I(t)), \quad t > 0, \; s(0) = s_0, \\ & I(t) = \int_0^{x_M(t)} \gamma(x, t, s(t)) u(x, t) dx, \quad t \geq 0, \end{aligned} \end{gather}\]

where \(x\) and \(t\) represent size and time, respectively, \(u(x, t)\) is the density function of consumers with size \(x\) at time \(t\), and \(s(t)\) is the quantity of available resource at time \(t\).
The model also introduces an undetermined size domain with a non fixed maximum size \(x_M (t)\) that changes with time, allowing a shrinking population.
This model is related with the work about the evolution of a population of *Daphnia magna* and the model which could represent it, introduced in [2] and also studied in [3].

We have proposed second order numerical methods which integrate the problem for a finite time interval \([0,T]\) in this PDE formulation.
We have analyzed its convergence to the solution when it is bounded, for a finite time integration, and we have performed numerical simulations based on a *Daphnia magna* population data.
Another approach to the model in the form of a system of two delays equations was introduced in [1] and the references therein.

We have also studied the use of the numerical scheme for a long time integration which shows us part of the dynamics of the model: there is a stable steady state that turns unstable when the parameter related with the carrying capacity increases.

This work is co-authored with Luis María Abia, Juan Carlos López-Marcos and Miguel Ángel López-Marcos.

Óscar Angulo is full professor of numerical analysis at the University of Valladolid (Spain).

Link to the Microsoft Teams meeting available at http://cdlab.uniud.it/events/seminar-20210701.

- [1] , Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example, J. Math. Biol., 61 (2010), pp. 277–318, DOI: 10.1007/s00285-009-0299-y.
- [2] , On the dynamics of chemically stressed populations: The deduction of population consequences from effects on individuals, Ecotox. Env. Safety, 8 (1984), pp. 254–274, DOI: 10.1016/0147-6513(84)90029-0.
- [3] , Numerical methods for structured population models: The escalator boxcar train, Numer. Methods Partial Differential Equations, 4 (1988), pp. 173–195, DOI: 10.1002/num.1690040303.