# The More The Merrier in The Math of Population Ecology

## Summary

This is a teaching module, directed to undergraduate students in applied mathematics, that derives a mathematical model of population ecology describing the role of dispersal in the survival of a population in danger of extinction. Students working with the module will write a computer code, using a software such as MATLAB or Mathematica, to obtain numerical solutions of the model. They will use the numerical simulations to explore the mathematical properties of the model and interpret the results in the ecological context

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## Learning Goals

The primary goals of this activity are twofold.

1. To increase student's motivation to learn mathematics by making it relevant to their lives; connecting mathematics learning to the goals and interests that students bring to college; and showing how mathematics relates to other disciplines, important civic questions, and technological challenges.

2. To develop basic skills in programming and scientific computing by writing their own computer code with graphical representation of the solutions.

## Context for Use

This module is directed to undergraduate students in applied mathematics, as in a course of mathematical modeling or calculus with differential equations. I have used versions of this material as an end of course group project, but it could also be used as individual project.

## Description and Teaching Materials

The More The Merrier Handout is a hands on introduction to mathematical modeling in population ecology. Population Ecology has ben defined as ``the science of what makes animal and plant populations change, persist, or go extinct"1.
A demographic tension between competition and cooperation drives the dynamics of many species. Population declines when competition among individuals for diminishing resources occurs at hight density levels. In this case fewer individuals benefit from more resources, but they may also suffer from a lack of support of conspecifics. Below a threshold value the death rate is higher than the birth rate because, on average, the individuals cannot reproduce successfully. This imposes a need for cooperation among individuals to increase the population and guarantee the survival of the species at low density.

Starting with the simplest model of exponential growth we progressively add enough features to the model in order to obtain a reasonable mathematical description of this phenomenon.

Students working with the module will write a computer code, using a software such as MATLAB or Mathematica, to obtain numerical solutions of the model. They will use the numerical simulations to explore the mathematical results in the context of this important topic of population ecology.

The file Population Script (private instructor-only file) is a MATLAB script that uses the ODE solver ode45 to obtain numerical solutions of the system of differential equations of the model, with corresponding graphical outputs. It calls the MATLAB function Population Vector Field (private instructor-only file) to compute the vector field of the ODE system. A Mathematica version for these computation is included in the Population Notebook (private instructor-only file.)

The More The Merrier Handout (Acrobat (PDF) 498kB Jan4 18)

1 B.W. Brook, "The Allure of the Few", PLoS Biol 6(5): e127 (2008). doi:10.1371/journal.pbio.0060127

## Teaching Notes and Tips

Although the module includes a self-contained introduction to the ecological motivations and derivation of the model, I usually dedicate a few lectures during the semester to discuss the material.

Basic skills in programming are expected to complete the project. These skills could be developed during the semester by integrating a software such as MATLAB or Mathematica as the main simulation tool within the course. Depending on the level of expertise required, instructors may or may not choose to provide the included private instructors-only MATLAB and Mathematica files to their students.

## Assessment

I use SENCER SALG pre- and post-assessment to assess the changes in student's perception of mathematics, its interaction with other disciplines, and its role in addressing relevant social issues. The SENCER SALG, http://salgsite.org/, is an online free course-evaluation tool that allows college-level instructors to gather learning-focused feedback from students. Instructors are guided through a wizard, using an adaptable template, to design their own instrument.

## References and Resources

V. Padron, M.C. Trevisan, Effect of aggregating behaviour on population recovery on a set of habitat islands, Math. Biosci. 165 (2000) 63.