Why Fit Models to Data?

“Why Fit Models to Data?” is a brief introductory chapter that sets the stage for the forthcoming chapters and serves as a link between Part 1, which contains the introductory chapters on mathematical ecology, and Part 2, which contains the statistical analyses of the models presented in Part 1: Part 1 illustrated how models can be used to make inferences about the real world, to help clarify ecological understanding, aid decision-making, and evaluate risk. However, if models are presented as hypotheses written in mathematical form, then it becomes possible to use statistical methods to determine which hypotheses have the greatest support. Part 2 will focus on developing statistical tools so that the reader will be able to express hypotheses as mathematical models, fit the models to data, and assess the degree of support for each. The chapter also illustrates the limitations of null-hypothesis testing in decision-making in high-dimensional, multicausal systems.

A Mathematical Model of Humanitarian Aid Agencies in Attritional Conflict Environments

Tools from mathematical ecology in a combat model with humanitarian aid agencies Conflict models have a long history of taking inspiration from mathematical ecology. In “A mathematical model of humanitarian aid agencies in attritional conflict environments,” McLennan-Smith et al. seek to enrich counterinsurgency (COIN) warfare models to account for modern and future complexities by incorporating nontrophic effects and the functional response from mathematical ecology. The authors consider the application of these ideas in a COIN scenario in which a humanitarian aid agency is present in the conflict environment to support the local population. In this scenario, the aid agency plays the unwilling role of a “hospital shield” whereby it is forced to, or inadvertently, shield combatants or weapons. In contrast to the typical behavior seen in the classic Lanchester system, this model gives rise to limit cycles and bifurcations that the authors interpret through a warfighting application. Finally, through a case study, the authors highlight the importance of the agility of an intervention force in achieving victory when humanitarian aid agencies are present.

On boundedness of solutions of quadratic differential equation sets

We obtain new sufficient conditions of asymptotic stability on solution cone of the set of usual uniform quadratic differential equations. For non-uniform quadratic sets (with non-zero linear part) we give the conditions of boundedness of solutions. We determine quadratic sets whose solution is bounded for any linear part and for any initial conditions. We provide examples of applications of the method in mathematical ecology.

Moments method in problems of optimal control of nonlinear models of logistic type

We investigate the problem of optimal control of two nonlinear models of mathematical ecology: logistic model and model of development in adversary environment. We consider four statements of optimal control problem, two criteria of quality (quadratic and linear ones). The solution is obtained either analytically, or numerically, by iterative approximations method.

Taylor collocation method for a system of nonlinear Volterra delay integro-differential equations with application to COVID-19 epidemic

The present paper deals with the numerical solution for a general form of a system of nonlinear Volterra delay integro-differential equations (VDIDEs). The main purpose of this work is to provide a current numerical method based on the use of continuous collocation Taylor polynomials for the numerical solution of nonlinear VDIDEs systems. It is shown that this method is convergent. Numerical results will be presented to prove the validity and effectiveness of this convergent algorithm. We apply two models to the COVID-19 epidemic in China and one for the Predator-Prey model in mathematical ecology.

CONTRIBUTION OF MODELING FOR A BETTER UNDERSTANDING OF MICROBIAL ECOSYSTEMS

This paper aims at presenting how a number of recent modeling approaches can be used for better understanding microbial ecosystems dynamics. In first part, an important question – the ability of certain ecosystems to exhibit overyielding – is investigated using a model-based approach. It is shown that classical competition theory cannot explain such phenomenon, thus invalidating a large class of classical mass-balance-based models, Rapaport et al. (2019). In second part, we show how new combinatorial approaches can be used to find the best combination of species of a functional ecosystem with limited complexity. More precisely classification approaches inspired from the work by Jaillard et al. (2018) are used and illustrated with simulations. Their robustness with respect to a number of experimental parameters (investigated in simulation) is studied. For ecosystems with higher richness, we show how another probabilistic approach proposed by Jaillard et al. (2014) may be useful. Keywords: Modeling, microbial ecosystems, microbial interactions, mathematical ecology, diversity, community assemblage.

Finite-dimensional maps describing the dynamics of a logistic equation with delay

This article discusses a family of maps that are used in the numerical simulation of a logistic equation with delay. This equation and presented maps are widely used in problems of mathematical ecology as models of the dynamics of populations. The paper compares the dynamic properties of the trajectories of these mappings and the original equation with delay. It is shown that the behavior of the solutions of maps can be quite complicated, while the logistic equation with delay has only a stable equilibrium state or cycle.