# Time-scale separation in Mathematical and Theoretical Biology

Università degli Studi Trento

38123 Povo (TN)

Tel +39 04 61/281508-1625-1701-3898-1980.

dept.math [at] unitn.it

**Luogo**: Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Sala Seminari "-1"

**Ore**: 10:00

**Relatore**:

- Bob W. Kooi (Vrije Universiteit Amsterdam)

**Abstract**:

Fast-slow systems in Ecology and Epidemiology are discussed. Predator-prey systems are of- ten modeled using ordinary differential equations: one for each trophic level. The slow-fast or time diversified system properties result from the assumption that the growth and loss rates of the preda- tor are much smaller than the growth rate of the prey. Two predator-prey model formulations will be studied. In the classical Rosenzweig-MacArthur (RM) model in absence of the predator the prey grows logistically. Consequently, no nutrient, resource for the prey, is modeled explicitly and the predator-prey model is described by a two-dimensional system. In this RM-model formulation the slow-fast assumption leads to an often-unrealistic assumption that the conversion efficiency needs to be small. In a mass balance (MB) chemostat model the nutrient is explicitly modeled leading to a three-dimensional system. Then for the fast-slow case the unrealistic assumption having a small efficiency is avoided. Because this model is based on mass conservation laws, by perfect aggre- gation the dimension of the system can be reduced by one leading to a two-dimensional system just as the RM-model.

We will show using singular perturbation theory [1,2], that the predicted long-term dynamics for the RM-model and bifurcation theory for the MB-model that they differ significantly when in both models the time scales at the two population levels differ. For instance, in the RM-model a so-called canard explosion occurs and not in the MB-model [1].

In [5] the role of vector modeling in a minimalistic epidemic model was studied. With the modeling of the vector population both ecological processes and disease transmission are important. In order to study the role of the host versus vector dynamics we combine simple host and vector models where the host model is a SIR model and the vector model an SI model [4]. When the rates of the processes involved in the vector dynamics are much larger than those in the host model we can use a time-scale argument to reduce the model. Then one

ordinary differential equation (ODE) is replaced by an algebraic equation. This is often implemented as a quasisteady-state assumption (QSSA) where the slow varying variable to set in equilibrium. Singular perturbation theory is a useful tool to do this derivation. An asymptotic expansion in the small parameter being the ratio of the two time scales for the dynamics of the host and vector, together with an invariant mani- fold equation gives the algebraic equation using symbolic analysis. In the case of a simplified SIS model for the host this algebraic

equation in its simplest form equal to the QSSA result and is just a hyperbolic relationship with saturation effects. This result is very similar to that of the Holling type II functional response in the case of the RM- and MB-model. In the case of a SIR model for the host, the situation is more complicated and will be discussed. See completed references in the attached poster.

**References**:

[1] B. W. Kooi and J-C. Poggiale, Modelling, singular perturbation and bifurcation analyses of bitrophic food chains,

Mathematical Bioscience, 301:93-110 2018.

[2] J-C. Poggiale, C. Aldebert, B. Girardot and B.W. Kooi, Analysis of a predator-prey model with specific time

scales : a geometrical approach proving the occurrence of canard solutions Journal of Mathematical Biology, 2019.

[3] M. Aguiar, S. Ballesteros, B.W. Kooi and N. Stollenwerk. The role of seasonality and import in a minimalistic

multi-strain dengue model capturing differences between primary and secondary infections: complex dynamics

and its implications for data analysis. Journal of Theoretical Biology, 289:181–196, 2011.

[4] F. Roacha, M. Aguiar, M. Souza and N. Stollenwerk. Time-scale separation and centre manifold analysis

describing vector-borne disease dynamics. International Journal of Computer Mathematics, 90(10):2105–2125,

2013.

[5] P. Rashkov, E. Venturino, M. Aguiar, N. Stollenwerk and B.W. Kooi. On the role of vector modeling in a

minimalistic epidemic model. Mathematical Biosciences and Engineering, 2019.

**Referente**: Maira Aguiar