## Mathematical Biology

The Mathematical Biology Research Group focusses on the development and analysis of models of population dynamics and evolution with applications in ecology, epidemiology and animal behaviour.

## Information Box Group

### Ben Bolker

Professor, Director of School of Computational Science and Engineering

**Research Area:** mathematical-biology

**Research Profile:** *mathematical-biology.*

I work on a wide variety of problems in theoretical ecology and evolution. Mathematical interests: the use of “moment equations” (equations for the mean, variance, and second-order spatial statistics such as spatial correlation) to understand the qualitative dynamics of biological communities (e.g. hosts and parasites, competing plant species)interacting in continuous spatial arenas. Biological interests: host/parasite interactions, broadly defined to include any “intimate association” among natural enemies and their victims: viruses, bacteria, and classical macroparasites. I work on the general dynamics of infectious disease epidemics, and particularly on the evolution of virulence. Statistical interests: fitting of parameters for ecological and evolutionary models, especially in a maximum likelihood settings. My recent work has focused on improvement and dissemination of tools for fitting generalized linear mixed models.

### Ben Bolker

**Professor, Director of School of Computational Science and Engineering**

### Jonathan Dushoff

Associate Member

**Research Area:** mathematical-biology

### David Earn

Professor

**Research Area:** applied-mathematics, mathematical-biology

**Research Profile:** *Epidemiology, Ecology, Evolutionary Game Theory*

Biological systems involve intricate interactions on many spatial and temporal scales. Mathematical models often make it possible to identify mechanisms behind complex biological processes and to predict the outcomes of environmental changes. My current research can be classified according to the time scale over which biological changes occur. Short time scales: Population dynamics of ecological and epidemiological systems. This work, which has implications for conservation of endangered species and eradication of infectious diseases, makes use of modern bifurcation theory. Analytical investigations of discrete maps and differential equations are usually supplemented by extensive numerical studies. Long time scales: Evolutionary dynamics of behavioural traits. This work, which is mainly based on game theoretical analysis, clarifies the adaptive significance of animal behaviour, ranging from cooperation and parental care to foraging and cannabilism.

### Gail S.K. Wolkowicz

Professor

**Research Area:** applied-mathematics, mathematical-biology

**Research Profile:** *Dynamical systems, bifurcation theory, population dynamics, mathematical ecology and epidemiology*

My students and I have been formulating and analyzing models motivated by questions in ecology and epidemiology. For example, one goal is to better understand basic population dynamics so that measurable criteria can be developed, enabling scientists to predict combinations of cultures of micro-organisms most effective and safest for use in such processes as water purification and biological waste decomposition. Other applications include pest control, the prevention of species’ extinction or the control or eradication of certain diseases. In order to elicit all the potential dynamics, a bifurcation theory approach is used so that the full spectrum of behaviour can be predicted for all appropriate parameter ranges and initial states. Computer simulations are used to elucidate complicated dynamics, to test conjectures, and to reveal properties of the models that are useful in developing analytic proofs. Symbolic computation is used to carry out complicated calculations. The analyses often lead to interesting abstract mathematical problems in dynamical systems, ordinary, integro- and functional differential equations, and bifurcation theory.

### Ben Bolker

Professor, Director of School of Computational Science and Engineering

**Research Area:** mathematical-biology

**Research Profile:** *mathematical-biology.*

I work on a wide variety of problems in theoretical ecology and evolution. Mathematical interests: the use of “moment equations” (equations for the mean, variance, and second-order spatial statistics such as spatial correlation) to understand the qualitative dynamics of biological communities (e.g. hosts and parasites, competing plant species)interacting in continuous spatial arenas. Biological interests: host/parasite interactions, broadly defined to include any “intimate association” among natural enemies and their victims: viruses, bacteria, and classical macroparasites. I work on the general dynamics of infectious disease epidemics, and particularly on the evolution of virulence. Statistical interests: fitting of parameters for ecological and evolutionary models, especially in a maximum likelihood settings. My recent work has focused on improvement and dissemination of tools for fitting generalized linear mixed models.

### Ben Bolker

Professor, Director of School of Computational Science and Engineering

**Research Area:** mathematical-biology

**Research Profile:** *mathematical-biology.*

I work on a wide variety of problems in theoretical ecology and evolution. Mathematical interests: the use of “moment equations” (equations for the mean, variance, and second-order spatial statistics such as spatial correlation) to understand the qualitative dynamics of biological communities (e.g. hosts and parasites, competing plant species)interacting in continuous spatial arenas. Biological interests: host/parasite interactions, broadly defined to include any “intimate association” among natural enemies and their victims: viruses, bacteria, and classical macroparasites. I work on the general dynamics of infectious disease epidemics, and particularly on the evolution of virulence. Statistical interests: fitting of parameters for ecological and evolutionary models, especially in a maximum likelihood settings. My recent work has focused on improvement and dissemination of tools for fitting generalized linear mixed models.

### Jonathan Dushoff

Associate Member

**Research Area:** mathematical-biology

### Jonathan Dushoff

Associate Member

**Research Area:** mathematical-biology

### David Earn

Professor

**Research Area:** applied-mathematics, mathematical-biology

**Research Profile:** *Epidemiology, Ecology, Evolutionary Game Theory*

Biological systems involve intricate interactions on many spatial and temporal scales. Mathematical models often make it possible to identify mechanisms behind complex biological processes and to predict the outcomes of environmental changes. My current research can be classified according to the time scale over which biological changes occur. Short time scales: Population dynamics of ecological and epidemiological systems. This work, which has implications for conservation of endangered species and eradication of infectious diseases, makes use of modern bifurcation theory. Analytical investigations of discrete maps and differential equations are usually supplemented by extensive numerical studies. Long time scales: Evolutionary dynamics of behavioural traits. This work, which is mainly based on game theoretical analysis, clarifies the adaptive significance of animal behaviour, ranging from cooperation and parental care to foraging and cannabilism.

### David Earn

Professor

**Research Area:** applied-mathematics, mathematical-biology

**Research Profile:** *Epidemiology, Ecology, Evolutionary Game Theory*

Biological systems involve intricate interactions on many spatial and temporal scales. Mathematical models often make it possible to identify mechanisms behind complex biological processes and to predict the outcomes of environmental changes. My current research can be classified according to the time scale over which biological changes occur. Short time scales: Population dynamics of ecological and epidemiological systems. This work, which has implications for conservation of endangered species and eradication of infectious diseases, makes use of modern bifurcation theory. Analytical investigations of discrete maps and differential equations are usually supplemented by extensive numerical studies. Long time scales: Evolutionary dynamics of behavioural traits. This work, which is mainly based on game theoretical analysis, clarifies the adaptive significance of animal behaviour, ranging from cooperation and parental care to foraging and cannabilism.

### Gail S.K. Wolkowicz

Professor

**Research Area:** applied-mathematics, mathematical-biology

**Research Profile:** *Dynamical systems, bifurcation theory, population dynamics, mathematical ecology and epidemiology*

My students and I have been formulating and analyzing models motivated by questions in ecology and epidemiology. For example, one goal is to better understand basic population dynamics so that measurable criteria can be developed, enabling scientists to predict combinations of cultures of micro-organisms most effective and safest for use in such processes as water purification and biological waste decomposition. Other applications include pest control, the prevention of species’ extinction or the control or eradication of certain diseases. In order to elicit all the potential dynamics, a bifurcation theory approach is used so that the full spectrum of behaviour can be predicted for all appropriate parameter ranges and initial states. Computer simulations are used to elucidate complicated dynamics, to test conjectures, and to reveal properties of the models that are useful in developing analytic proofs. Symbolic computation is used to carry out complicated calculations. The analyses often lead to interesting abstract mathematical problems in dynamical systems, ordinary, integro- and functional differential equations, and bifurcation theory.

### Gail S.K. Wolkowicz

Professor

**Research Area:** applied-mathematics, mathematical-biology

**Research Profile:** *Dynamical systems, bifurcation theory, population dynamics, mathematical ecology and epidemiology*

My students and I have been formulating and analyzing models motivated by questions in ecology and epidemiology. For example, one goal is to better understand basic population dynamics so that measurable criteria can be developed, enabling scientists to predict combinations of cultures of micro-organisms most effective and safest for use in such processes as water purification and biological waste decomposition. Other applications include pest control, the prevention of species’ extinction or the control or eradication of certain diseases. In order to elicit all the potential dynamics, a bifurcation theory approach is used so that the full spectrum of behaviour can be predicted for all appropriate parameter ranges and initial states. Computer simulations are used to elucidate complicated dynamics, to test conjectures, and to reveal properties of the models that are useful in developing analytic proofs. Symbolic computation is used to carry out complicated calculations. The analyses often lead to interesting abstract mathematical problems in dynamical systems, ordinary, integro- and functional differential equations, and bifurcation theory.