Mathematical biology

A/A 2009-10

Short summary of topics presented in class

References to text-books:

[BCC] F. Brauer and C. Castillo-Chavez, Mathematical models in population biology and epidemiology, Springer 2001

[Br] N. Britton, Essential mathematical biology, Springer,2003

- 15/2

o Models of Malthus and Verhulst: [Par. 1.1-1.2 of [BCC]]

§ Assumptions: homogeneous population (no differences of age, sex, spatial location, genetics, health...); large population (deterministic, not stochastic model).

§ Fertility and mortality rates. Malthus's equation and solution.

§ Verhulst (logistic) equation. Solution. Interpretation of parameters: r (intrinsic growth rate) and K (carrying capacity).

- 18/2

o Exponential distribution of length of sojourn in each stage; [Par. 1.7 of [BCC]]

o Non-dimensionalizing method applied to the logistic;

o Qualitative analysis of generalized logistic model:
N' - N g(N) with g(N) > 0 for 0<N<K and g(N) < 0 for N >K. [Par. 1.4 of [BCC]]

- 19/2

o General structure of models for the growth of a single homogeneous species, based on ordinary differential equations: compensation, depensation, critical depensation (or Allee effect). [Par. 1.4 of [BCC]]

o Stable and unstable equilibria.

o The Spruce Budworm model: motivation for the equation; search for equilibria by plotting on the same graph r(1-x/k) and Bx/(A+x²) : equilibria correspond to intersection of the two graphs; idea (no rigorous computation) that there will be one positive equilibrium for B close to 0 and B large, and three for B intermediate (two in critical cases); bifurcation graph (equilibria against B); suggestion that there can be catastrophic transition as B changes. [This will be discussed more in detail further in the course] [Par. 1.8 of [BCC]]

o Harvesting a single population: constant yield and constant effort strategies. Equilibria with each of the strategy. Posing the problem of finding optimal effort to have maximum sustainable yield [Par. 1.5 of [BCC]]

- 22/2

o Volterra predator-prey system [Par. 4.1 of [BCC]]

§ Assumptions, isoclines and general structure of the vector field.

§ Existence of a prime integral; consequently, all solutions are periodic.

§ The averages over a period are equal to the equilibrium values.

§ Volterra's principle: how the balance between prey and predators is modified by fishing (hunting, harvesting...)

o The chemostat [Par. 4.2 of [BCC]]

§ Description of the apparatus, and basic model

§ Non-dimensionalizing the equations

- 26/2

o The chemostat (conclusion)

§ u+v = 1 is an invariant and attractive set

§ Geometrically, this (together with the structure of the vector field) implies that all solutions converge to the unique equilibrium

§ Reduced one-dimensional equation on u+v = 1.

o Reminders on linear autonomous systems y' = Ay. Solution through eigenvalues and (generalized) eigenvectors. [Par. 4.4 of [BCC]; see also any book on ordinary differential equations]

§ Condition on trace and determinant for all the eigenvalues of a 2x2 matrix to have negative real part.

o Stability of equilibria of nonlinear systems: [Par. 4.3 of [BCC]; see also any book on ordinary differential equations]

§ Linearization theorem (Liapunov's theorem).

o Volterra prey-predator system with logistic growth:

§ Equilibria and their stability through linearization. [Par. 5.2 of [BCC]: first example]

- 1/3

o Prey-predator systems of Rosenzweig-MacArthur type

§ Functional and numerical response: functional responses of 1st, 2nd and 3rd type. Holling's functional response.

§ Analysis of equilibria and their stability. Condition for the instability of the positive (i.e. with both components strictly positive) equilibrium. [Par. 5.2 of [BCC]]

- 4/3

o Discussion of exercises in class

- 5/3

o Some properties of solutions of autonomous differential equations: see most books on ordinary differential equations/dynamical systems; my notes (in Italian) on the Web page]

§ [Positively] invariant sets

§ w-limit sets and their properties

§ Liapunov functions and Liapunov-LaSalle theorem

o Use of Liapunov-LaSalle theorem to prove global stability of positive equilibrium in Volterra prey-predator system with logistic growth [see my notes on the Web page]

- 8/3

o Poincaré-Bendixson theory for planar systems [a short summary in Par. 4.4 of [BCC]; theorems and proofs e.g. in Wiggins or Perko ]

- 12/3

o Criteria of Bendixson and Bendixson-DuLac for the non-existence of periodic solutions [Par. 4.4 of [BCC] or the cited books];

o Use of the criteria for proving the global asymptotic stability of positive equilibrium in Volterra prey-predator system with generalized logistic growth;

o Use of Poincaré-Bendixson theory to prove the existence of at least one periodic solution of Rosenzweig-MacArthur prey-predator systems when Y'(H*) > 0.

o Holling's prey-predator system: non-dimensionalization; construction of a bifurcation graph showing stable and unstable equilibria and periodic solutions.

- 15/3

o Rosenzweig-MacArthur prey-predator systems: construction of a compact positively invariant set including all equilibria.

o Holling's prey-predator system: proof of the global stability of the positive equilibrium via a suitable DuLac function. [see my notes on the Web]

o Computer simulations:

§ Holling's prey-predator system for different parameter values.

§ A prey-predator system by Hofbauer-So, showing (for certain parameter values) multiple periodic solutions.

- 18/3

o Lotka-Volterra competition [Par. 5.1 of [BCC] ]

- 19/3

o Monotonicity of solution of 2-dimensional cooperative or competitive systems [my notes; check also last part of Par. 5.1 of [BCC] ]

o Competition for one non-renewal resource: competitive exclusion [my notes]

- 22/3

o Chemostat model with 2 species: competitive exclusion [my notes]

o Competition between 2 predators for 1 logistic prey: equilibria and their stability. [my forthcoming notes; see also Par. 5.11 of [BCC] ]

- 26/3

o Competition between 2 predators for 1 logistic prey: conditions for the stability of the boundary periodic orbit; there exist parameter values such that all boundary attractors are unstable: intuitively, there will exist an internal boundary attractor.

o Simulations of the system showing a positive periodic orbit.

- 29/3

o May-Leonard model for non-transitive competition. Conditions for the stability of the internal (symmetric) equilibrium. Proof of the global stability of the internal equilibrium or of the boundary heteroclinic cycle. [my notes]

o Definition of [uniform] persistence. [an idea in Par. 5.9 of [BCC] ]

- 8/4

o Logistic equation with delay. Motivation. Stability of the equilibrium: statement of the method of linearization and exponential solutions; values of delay that allow for imaginary solutions of the characteristic equation; conditions under which K is asymptotically stable or unstable. [Par. 3.2 of [BCC] ]

- 9/4

o Idea of bifurcation theory. Statement and illustration of Hopf bifurcation theorem for ODEs [see my notes]; without details, statement that the theorem would hold also for delay differential equations and more complex equations. Prototype example: r'=ar±r³, q'=1.

o Application of the logistic equation with delay to Nicholson's blow-flies. [Par. 3.6 of [BCC] ]

- 12/4

o Distributed delay with exponential or gamma kernels: reduction to a system of ODE. [Par. 3.4 of [BCC] but not using the reduction to ODEs]

o Routh-Hurwitz conditions for the eigenvalues of a 3x3 matrix. Application to the stability or instability of the equilibrium K for the logistic with gamma-distributed delay

o Discrete maps as models for populations living with discrete growing seasons and no survival between them (annual insects or plants).
Examples: Malthus discrete, logistic discrete, Ricker's model, Beverton-Holt model. equilibria.
Condition for equilibrium stability. The positive equilibrium of Ricker's model is stable for 1<a<e², unstable for a>e². Solutions of period 2 as fixed points of the second iterate. Statement (without details) of period-doubling bifurcation. Hints to "chaotic" solutions for larger a.
[Par. 2.1-4 of [BCC] ]

- 15/4

o Models with discrete time and age classes. Leslie matrices. Euler-Lotka equation. Eigenvalues of Leslie matrices: proof of the existence of a (strictly) dominant positive eigenvalue, under a condition on fertile age-classes. [Par. 8.1 of [BCC] and notes on the Web page]

- 16/4

o Leslie matrices: long-tem stationary age distribution.

o General linear matrix models. Main results of Perron-Frobenius theory [notes on the Web page] and their application to linear matrix models.

o Nonlinear models with discrete time and age classes. Equilibria. Conditions for stability.

- 19/4

o Molecular networks: main ideas, motifs... [slides on the Web page; see also Par. 6.3 of [Br]]

- 23/4

o Cell cycle: basics and models for its regulation [slides on the Web page]

- 26/4

o A model for enzyme kinetics. Tikhonov's theorem: statement and application to enzyme kinetics: derivation of Michaelis-Menten equation. [Par. 6.2 of [Br]]

o Application of Tikhonov's theorem to FitzHugh-Nagumo-like equations: dynamics of excitation.

- 29/4

o Exercises

- 30/4

o Relaxation oscillations in fast-slow systems: semi-intuitive derivation through Tikhonov's theorem. [see Par. 6.4 of [Br]]

o Summary of main empirical facts about impulse transmission in axons. [see notes on the Web page for impulse transmission in neurons]

o Hodgkin-Huxley model for the giant axon of squid; equations in spatial clamp.

§ Main properties of the solutions.

- 3/5

o Identification of fast and slow variables in Hodgkin-Huxley model

o Phenomenological modelling via FitzHugh-Nagumo model

§ Properties of the solutions

§ Possibility of periodic solutions (relaxation oscillations) in FitzHugh-Nagumo with a constant input

- 7/5

o Presentation (without justification) of the partial differential equation for V(t,x) in Hodgkin-Huxley model in an axon of finite length

o Concept of a travelling wave solution for a reaction-diffusion system

o FitzHugh-Nagumo equations in time and space

§ Intuitive description of the properties of a travelling wave solution in the limit of separation of different scales in the equations for V and w

o Nagumo equation with diffusion

§ Travelling wave solutions connecting 0 and 1

· sketch of the proof of the existence of a unique wave speed;

· an explicit travelling wave solution.

- 10/5

o Stochastic models for single populations: idea of a birth-and-death processes [a reference for this part is H.M. Taylor and S. Karlin, An introduction to stochastic modeling, Academic Press, 1994; see notes on the Web]

o Summary of main ideas about Markov processes with discrete state space:

§ infinitesimal transition rates.

§ Kolmogorov differential equations.

o Sketch of how to build a Markov process from the infinitesimal transition rates:

§ embedded jump Markov chain

§ exponential waiting times

- 14/5

o Birth-and-death processes

o Method of generating function to obtain explicit solutions in the linear case (sketch)

o Probability of extinction:

§ reduction to computations through the embedded jump Markov chain;

· in the linear case, this is the problem of gambler's ruin: explicit solution;

· semi-intuitive explanation that extinction is certain in a logistic-type case.

o Mean time to extinction:

§ linear system to find them (minimal nonnegative solution)

§ explicit computation in the linear case when l£m.

- 17/5

o Mean time to extinction in a simple logistic case: linear up to size K, then birth rate equal 0.

§ Asymptotic (for large K) estimate for it. A numerical illustration.

o Idea of stationary and quasi-stationary probabilities.

o SIR epidemic model in a closed population.

§ Threshold result.

- 20/5

o Exercises

- 21/5

o SIR epidemic model in a closed population:

§ Interpretation of R0

§ Equation satisfied by S(+∞)

§ Peak value of infectives

o SIR epidemic model in a population with demography

§ Equilibria and their stability

§ Threshold result

§ Global convergence to equilibrium

§ Quasi-period of damped oscillations to the endemic equilibrium

- 24/5

o SIR stochastic model in a closed population:

§ infinitesimal transition rates;

§ threshold result as population size goes to ∞.