Final Project Descriptions Introduction to Mathematical Biology Professor: Paul J. Atzberger. Project I: Predator-Prey Equations

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1 Final Project Descriptions Introduction to Mathematical Biology Professor: Paul J. Atzberger Project I: Predator-Prey Equations The Lotka-Volterra Predator-Prey Model is given by: du dv = αu βuv = ρβuv γv where αu models the birth of the prey, βuv models death of the prey by being captured by predators, ρβuv models the rate of birth of the predators (which depends on how many prey are captured), and γv models the death of the predators. The basic model exhibits solutions which have periodic oscillating predator and prey populations. There is one non-trivial steady-state which is a center which is stable, but is not asymptotically stable. Consequently, the initial conditions for the predator and prey population size play an important role in determining the overall dynamics. To obtain the basic Lotka-Volterra Predator-Prey Model a number of assumptions were made. In particular, the predators are assumed to die at a constant per capita rate and the prey are assumed to reproduce at a constant per capita rate. The predator birth rate is assumed to depend only linearly on the rate at which predators catch prey. While the basic model may be a good starting point in modeling predator-prey populations, it neglects many features of populations which may be potentially important in practice. For example, the effect of the different seasons of the year on the birth rate of the prey and the death rate of the predators was not taken into account. For instance, in the winter we might expect the birth rate of the prey population to be greatly reduced and the death rate of the predators to increase. An interesting question is how these periodic variations effect the population dynamics, which is already prone to its own intrinsic periodic oscillations through the predator-prey interactions. In this project you will explore modifications of the basic predator-prey model using both analytical and computational techniques. One way to incorporate seasonality into the model is to modify the constant rates above, say β, to be β(1 + ǫ sin(2πωt)) where ǫ controls the influence of the seasonality and ω controls the frequency of the seasons. Use matlab codes to simulate the model and explore the influence of seasonality separately for each rate constant, then try combinations. In your report carefully address the following questions: (i) How does the dynamics of the system change qualitatively when seasonality is incorporated into the various rate constants? (ii) Which rate constants have the most dramatic effect on the dynamics when modified to incorporate seasonality? (Birth of prey, death of predators, rate of predation, or are they all comparable?)

2 (iii) What influence does ω have on the effects of the seasonality? If ω is very large is this any different than when ω is small? What about the effect of ǫ small verse very large? Be careful to keep ǫ small enough so that all rates remain positive. (iv) Are there steady-states or limit cycles for the system? If so, try to determine their stability either by linearization or by arguing in terms of phase-plane portrait. Give a biological interpretation of why the steady-states or limit cycles occur and how they are stabilised. On the class website you can find numerical codes for computing trajectories of the system and for the computation of Poincare maps.

3 Project II: Ecological Modeling Species of plankton grow on the surface of the oceans and play a fundamental role in the oceanic food chain. They can be classified into two broad categories, phytoplankton and zooplankton. Phytoplankton are plant species that use photosynthesis to sustain themselves and are often limited in their growth by available nutrients in the environment. Zooplankton are animal species and feed on the phytoplankton. In this project you will explore the effects of the nutrients in the environment and the interaction of the two species with one another. A basic model for the plankton populations and nutrients is the following: dp dz dn = cnp dpz ap = dpz bz = ap + bz cnp where P, Z model the phytoplanton and zooplankton population sizes and N models the amount of the available nutrient nitrogen. The ap, bz terms model the rate of nitrogen production which occurs when plankton die and cnp models the rate of consumption of the nitrogen by phytoplanton. The terms appearing in the equations for P and Z follow the usual predator-prey interpretation (see Project I). Some species of phytoplankton form massive colonies of cocolithiphore clusters on the surface of the ocean and produce the gas dimethyl sulfide which is thought to play an important role in controlling the buoyancy of the clusters. When dimethyl sulfide is released from the ocean surface into the atmosphere it becomes a good base for condensation of water and the formation of clouds. One hypotheses is that the phytoplankton, which depend on sunlight for photosynthesis, could in principle form a regulating loop with the climate through the role dimethyl sulfide plays in cloud formation. In this project the hypothesized feedback loop and its effects on the dynamics of the plankton populations will be explored. In the model the reproduction of the phytoplankton was modeled as depending on the size of the phytoplanton population and the amount of the nutrient nitrogen with a constant coefficient c. Since the rate of reproduction of the phytoplanton population is expected to depend on the amount of sunlight available for photosynthesis we can incorporate effects of cloud cover by using a non-constant coefficient c. One simple model is to modify the photosynthetic rate constant c to c(1 Q), where Q is the average density of clouds in the sky having a range from 0 for no clouds to 1 for an overcast sky saturated with dark clouds. To model how the clouds depend on the phytoplankton population we could use a logistic equation for 1 Q with an effective carrying capacity of 1. This yields the model equations P for Q: dq = r(1 Q)(1 P(1 Q)). In the project use the analytic methods discussed in class and the matlab codes to address the following questions:

4 (i) Does the dynamics change significantly when the cloud-phytoplankton feedback loop is incorporated into the model? (ii) If we introduce a delay τ into the equation for the cloud formation so that P(t) is replaced by P(t τ) how does this change the dynamics? (iii) What are the steady-states or limit-cycles for the model with and without delays? Which are stable? Argue using linearization theory and plots of the dynamical trajectories. (iv) What happens in the model if the cloud formation is influenced by the seasons? In particular, what happens if the seasons amplify or diminish the effects of the phytoplankton on cloud formation in the sense that P is modified to (1+ǫ sin(2πωt))p. For example, what happens when ω is small? When it is large? On the class website can be found numerical codes to solve the ODEs.

5 Project III: Epidemiology of Diseases The spread of many diseases can be modeled by classifying individuals in a population into one of three classes. An individual can either be disease-free but susceptible (S), infected with the disease and contagious (I), or immune to the disease and recovered (R). Many features of how the disease behaves in the population can be described in terms of how the size of these classes changes over time and by the steady-states of the system. We showed in class how to predict whether a disease will lead to an epidemic and how to estimate the total number of individuals infected over the course of the epidemic. We also discussed how to predict whether a disease will be endemic and how to estimate the fraction of the population that will be chronically infected. The effectiveness of various vaccination strategies was also analyzed. In many such disease models the law of mass action is used to model the interactions between susceptible and infected members of the population. This is based on the underlying assumption that the classes are well mixed so that the members of each of the classes are uniformly spread-out over the regions occupied by the population. For many diseases this assumption is unrealistic and in fact does not give good results, for example, some diseases depend importantly on social interactions and lifestyle habits which play a central role in transmission. In this project you will explore the role that non-uniform interactions play in the dynamics of diseases. One approach to modeling non-uniform interactions of the population is to use a graph (social network) to model the interactions between individuals. The individuals are modeled as the nodes of a graph where an edge is formed between any pair of individuals who are in contact. The population members then interact using the specific structure of the graph. To model the network of interactions we shall use a random graph where an edge is formed between a pair of individuals with probability 0 p 1. Using the random graph one infected individual is introduced into the population. Over the time interval δτ the following cycle of events occur to update the population. (i) Each individual who is connected to an infected member has probability βδτ of becoming infected. (ii) Infected individuals have a probability of γδτ of recovering from the disease over the time step. We exclude the newly infected individuals from consideration in this latter step. Using matlab codes and the analytic techniques discuss in class explore features of this model. In your report the following questions should be addressed: (i) What are the sizes of the S, I, R classes as the disease spreads throughout the population? How does the parameter p influence the occurrence of and time course of the epidemic? Is there a critical value of p? Is there an endemic state? What is its size? (ii) How does the random network model compare to the S, I, R models discussed in class when the same parameters are chosen for β,γ, etc..? Does the size N of the population play a role? What happens when N is small? What appears to be the range of values for p that gives good agreement with the S,I,R models in class?

6 (iii) How does the model change qualitatively when disease-unrelated death and disease-related death are added to the model? Does the model agree well with the results derived in class? What range of p leads to good agreement? For what values of p does the model behave differently? On the class website can be found codes to generate the random graphs and to perform the updates for the dynamics of the model.

7 Project IV: Neuroscience The Hodgkin-Huxley-Katz equations originally derived for experiments on giant squid axons provides a mathematical model for how electrical signals are propagated in many neural tissues. The equations model how individual neurons respond to a stimulating depolarising pulse of current and how the subsequent action potential is propagated along the axon. The propagation of the action potential occurs in four phases: (i) a positive feedback phase where the depolarisation is increased by the opening of sodium ion channels, (ii) an excited phase of reversed polarisation where potassium ion channels are opened to counterbalance the reverse polarisation, (iii) a hyperpolarised phase where the sodium ion channels are closed and the neuron begins its recovery to the rest-state, (iv) a refractory period where the rest-polarisation has been substantially recovered but the sodium channels remain inactive. In this project you will explore how a neuron responds to various types of stimuli using a variant of the Hodgkin-Huxley-Katz model. To make the problem more tractable we shall work with a reduced system of equations derived by FitzHugh and Nagumo which model only two variables, an effective conductance and voltage associated with an action potential. The FitzHugh-Nagumo equations are: ǫ dv dw = f(v,w) + I(t) = g(v, w) where f(v,w) = v(v a)(1 v) w, g(v,w) = v c bw, and the value of ǫ > 0 is typically small. The function I(t) represents the input stimuli to the neuron. Using matlab codes and the analytic techniques discussed in class you will explore the response of the neuron for various types of stimulating currents I(t). In your report address the following questions: (i) What does the time series for the voltage look like when the input stimuli is a single pulse of current? What magnitude and duration is required to get a response from the neuron? Analyze this in terms of the phase-plane of the equations. (ii) What does the time series for the voltage look like when the input stimuli is a constant current? How does this depend on the magnitude of the current? Is there a critical magnitude? How does the constant current change the phase-plane of the system? (iii) How does the time series change if we make the input current a sequence of pulses of fixed duration? What types of behavior does the model exhibit as the frequency and duration of the pulses are varied? How might this behavior arise? Try to explain it in terms of the phase-plane. (iv) What happens if we make the input current white noise? How does the behavior depend on the magnitude of the noise? What if we make the stimulating currents pulsed again but with an exponential waiting time between pulses? Give an explanation of what happens in terms of trajectories in the phase-plane.

8 On the class website can be found numerical codes to integrate the FitzHugh-Nugamo equations and to generate the currents I(t).