Mathematical Biology

Continuous-time Models

Population dynamics that occur continuously in time

Ordinary differential equations

The Malthusian Model

• Describe and predict the growth of the human population in England

• N′(t) = bN − dN = (b − d)N = rN,

The Verhulst Model

• Growth of a population is assumed to slow down as the population becomes larger due to crowding effects

• N’(t) = r(1-(N(t)/K))

Steps of nondimensionalisation

1. Determine units of all parameters

2. Rewrite dependent and independent variables as nondimensional quantities

3. Find suitable choices for the undetermined constant

Steady State

• Also referred to as equilibria of the system

• Values N = N* that satisfy f(N*) = 0.

• States the system cannot leave -N′(t) = 0 if N = N*

Types of steady states

• stable if f’(N*) < 0

• unstable if f’(N*) > 0

• inconclusive if f’(N*) = 0

Hysterisis

• history-dependence

• The state a system is in not only depends on the current environmental conditions but also on which states the system has been in previously

Tipping point

A small environmental change leads to large changes in an ecosystem that cannot be reversed

Delay-differential equation

• Accounts for delays in a system

• dN(t)/dt = f (N(t), N(t − T))

Discrete-time models

• Used to describe biological systems in which the time-continuous nature of ODE models is inappropriate

Difference equations

N_t+1 = f(N_t)

N_t+1 = rN_t , r ≥ 0.

Steady state of difference equations / types

• N* = f(N*)

• Linearly stable if |f’(N*)| > 1

• Linearly unstable if |f’(N*)| < 1

Types of (un)stability for discrete-time models

• oscillatory stable if −1 < f′(N*) < 0

• monotonically stable if 0 < f′(N*) < 1

• oscillatory unstable if f ′(N*) < −1

• monotonically unstable if f ′(N*) > 1

Cobweb diagram

• Graphically determine steady states, stability and other solution behaviour

Enzyme reaction

enzymes combine with a substrate to form a complex that is subsequently degraded into products and the enzymes

Catalysts

they accelerate or inhibit a chemical reaction but are not degraded by doing so

The Law of Mass action

The rate of a biochemical reaction is proportional to the product of the concentrations of the reactants

Singularly perturbated problem

• Problems in which “the most important term” is multiplied by a small constant ε ≪ 1

• The implication is significant, setting ε = 0 does not provide an appropriate approximation

Method of matched asymptotic

Split the system into two simpler subsystems, analyse both and combine the results in a meaningful way.

Oscillations

Not all reactions lead to a gradual decay of a substrate and gradual increase of a product

Periodic solutions = limit cycles

The Poincaré-Bendixon Theorem

if there is a confined set in the system’s phase space that contains only a single steady state, and if that steady state is unstable, then solutions converge to a limit cycle

Confined set

A subset of the phase space that solution trajectories cannot leave

Stability of Steady states

If both eigenvalues λi ∈ C, i = 1, 2 of the Jacobian evaluated at the steady state (x, y) = (x^∗,y^∗) have negative real part, maxi=1,2 Re(λi) < 0, then the (x^∗,y^∗) is stable. If at least one eigenvalue has positive real part, maxi=1,2 Re(λi) > 0, then the steady state is unstable.

Hartman-Grobman Theorem

trajectories in a nonlinear system close to a steady state are well-approximated by trajectories of the corresponding linearised (about the same steady state) system, provided eigenvalues have non-zero real part.

Competitive interaction

occur when one population reduces the net growth of the other and vice versa

Interference competition

Direct competition for resources (e.g. fighting)

Uses the competitive Lotka-Volterra model.

Competitive Exclusion Principle

Coexistence of species is possible if every species exerts a higher competitive pressure on individuals in its own population than on individuals in other populations

(intraspecific competition among each species is stronger than the interspecific competition it exerts on its competitor)

Exploitation competition

occurs when species compete for a single limiting nutrient.

Tilman’s R^* rule

The species that reduces the resource density to the lowest level at steady state outcompetes the other

Cooperation/ symbiosis

• Occurs when one population promotes the growth of another and vice versa.

• For example, birds that feed on berries help seed dispersal. Thus, the interaction is beneficial for both populations.

Predator-prey and host-parasite interactions

• One population (the predator or parasite) benefits from the interactions with the other species (the prey or host).

• The prey species, in turn, is harmed by these interactions.

• Often feature oscillations with a slight lag

Jury Conditions

The eigenvalues λ1, λ2 ∈ C of the matrix J ∈ R^2×2 satisfy |λ1| < 1 and |λ2| < 1, if and only if det(J) < 1, and 1 + det(J) > |tr(J)|

Compartmental models

The total population (humans, animals, plants) is split into different compartments, and over time, individuals can “move” around the compartments

SIR Model and Assumptions

• three compartments: susceptibles (S), infected (I), and recovered (R)

• S can become infected through contact with I, I can recover, and R immune from the infection.

• assume that no birth and death occurs in the population

• that immunity of recovered individual does not wane

Force of infection

The rate at which susceptibles become infected is λ = βI in this SIR model

Basic Reproduction number

R0 describes the average number of secondary infections caused by a single infected individual in an otherwise susceptible population

R0 > 1 leads to an epidemic 

Extensions to the SIR model

• Latent (L): individuals who have been exposed to the disease but are not yet infectious

• Quarantined (Q): infected individuals that are subject to quarantine measures

• Chronic (C): chronically ill individuals

• Diseased (D): individuals can die

• Vaccinated (V): individuals who are vaccinated

• Individuals with maternally-derived immunity (M): offspring is initially immune to disease due to maternal antibodies

Vaccination and the SIR model

• Assume that a proportion 0 ≤ p ≤ 1 of the population receives a vaccine and that this vaccine provides immunity for all recipients.

• vaccination program prevents an epidemic provided that R₀^vacc < 1 ⇔ p > 1 − γ/βN = 1 − 1/R0 .

Vector-borne diseases

• Many diseases are caused by parasites that spend periods of their life cycle in another organism, typically referred to as a vector

• They can enter the blood stream of a human due to a bite from an infected mosquito

• In turn, mosquitos become infected by biting an infected human. Thus, human-to-human infection occurs via mosquitos (vector)

Assumptions of vector-borne model

• No recovered class → no immunity

• disease dynamics are much faster than the human population dynamics and thus disregard any birth or death processes.

• vectors are assumed to reproduce and die at same rate all newborn vectors are susceptible

2 Vector-borne steady states

• disease-free steady state (iH, iV ) = (0, 0)

• endemic steady state (iH, iV ) = (i∗H, i∗V ), where i∗H = (αHαV − 1)/ (αV(αH + 1)), i∗V = (αHαV − 1)/ αH(αV + 1)

Stability of vector-borne

• (iH, iV ) = (0, 0) is stable if and only if αHαV < 1.

• endemic steady state (iH, iV) = (i^∗H, i^∗V) is stable if and only if αHαV > 1

Ecological models

Provide information on the dynamics of (interacting) populations with pre-defined properties do not change over time.

Do not count for evolution.

Adaptive dynamics

A method that allows us to describe how model parameters evolve over time. Its main idea is to determine parameter values that represent a possible endpoint of evolution.

Evolutionary and Convergence stable

A strategy that cannot be invaded by other strategies

The eco-evolutionary dynamics can converge to it.

Fitness gradient

if ∂g/∂m |m=r

• > 0 then the trait evolves towards larger values

• < 0 then the trait evolves towards smaller values.

Evolutionary equilibrium

Whenever the fitness gradient is zero there is no directional selection in either direction occurring.

Can be stable (fitness maximum) if ∂²g/ ∂m²|m=r=r ∗ < 0, or unstable (fitness minimum) if ∂²g/ ∂m²|m=r=r ∗ > 0.

Pairwise Invasion plot

• Plot the sign of the invasion fitness g(m; r) against r and m, with the resident trait r on the x-axis and the mutant trait m on the y-axis.

• Plotted in 2d, with regions of g(m; r) > 0 and g(m; r) < 0 being distinguished through different shadings

Evolutionary stable on PIP

• Evolutionary equilibria occur for trait values at which drawing a vertical line onto the PIP does not cause a change of sign(g(m; r)) as it passes through the diagonal m = r.

• If sign(g(m; r)) ≤ 0 along the vertical line, then the evolutionary equilibrium is stable

• If sign(g(m; r)) ≥ 0 along the vertical line, then the evolutionary equilibrium is unstable

Requirement for convergence stable

Close to the evolutionary equilibrium, directional selection for c < r∗ is positive and directional selection for c > r∗ is negative.

Convergence stable on PIP

If, for r = m < r∗ the invasion fitness is positive above the diagonal and negative below, and if for r = m > r∗ the invasion fitness is negative above the diagonal and positive below, then the evolutionary equilibrium r ∗ is convergence stable, and otherwise unstable.

Outcomes of evolution model

• evolutionary stable and convergence stable - equilibrium presents a possible outcome of evolution

• evolutionary stable and convergence unstable - no influence on long-term outcome

• evolutionary unstable and convergence stable - evolutionary branching point

• evolutionary unstable and convergence unstable - no influence on long-term outcome

Spatio-temporal dynamics

• Consider both space and time

• Can capture clusters of individuals

• This leads to partial differential equations (PDEs) or integrodifference equations

Conservation equation

One-dimensional conservation eqn

The flux that changes the total population in V is that entering through the cross-section at x and leaving through the cross-section at x+ ∆x.

Fickian Diffusion

The random motion of individual molecules (random walk)

Nonlinear diffusion

diffusion parameter depends on population density (e.g. overcrowding)

Convection or Advection

Directed movement (e.g. pollen movement in a particular wind direction)

Taxi

Directed movement in response to an external chemical or physical signal, for example bacteria sensing and moving towards a food source.

Chemotaxi, Haptotaxi, Thermotaxi

• movement directed by a chemical gradient

• movement directed by a gradient in adhesive substances (often bacteria moving in response to extracellular matrix elements)

• movement directed by a gradient of temperature

Infinite domain

• the density is not influenced by the boundary

• used when systems considered are so large that it can be safely assumed that dynamics in large parts of the system are unaffected by the boundary – for example vegetation dynamics in the desert.

Periodic BC

• model densities are periodic functions

• used when, for example, particles move on a torus, or also to mimic infinite domains in numerical simulations.

Dirichlet BC

• density (concentration) is fixed at the boundary (but can still be a function of time and space)

• For example, consider two reservoirs of chemical reactants placed at the ends of the domain, that are held at densities (concentrations) c1(t) and c2(t), respectively.

No-flux BCs

Particles cannot escape from the domain

Non-homogeneous Neumann BC

Flux on the boundary is given by some non-zero function.

Mixed Robin BC

A combination of Dirichlet and Neumann BCs. (Can be homogeneous or non-homogeneous)

Biological Waves

• A quantity of interest (population density, chemical concentration, etc.) spreads throughout space over time.

• e.g. spread of grey squirrels throughout UK

Travelling wave

If x ∈ R denotes space and t ∈ R denotes time, a function u(x, t) is a ____ if it can be expressed as

u(x, t) = u(z), z = x − vt, v ≠ 0

Pulse, Front, Periodic

• Travelling ___: u(x, t) → a, as x → ±∞

• Travelling ___: u(x, t) → a, as x → −∞, u(t, x) → b, as x → +∞ and a ≠ b.

• ___travelling wave: u(x, t) is a travelling wave that is a periodic function in x-

Fisher’s Equation

• one-dimensional reaction-diffusion equation comprising a logistic growth term and standard diffusion

• apply to any setting in which a population follows approximately logistic growth and moves randomly.

Bendixson-Dulac Theorem

If there exists a function φ(W, P), with φ ∈ C 1 (R 2 ), such that ∂(φF)/∂W + ∂(φG)/∂P has the same sign (≠ 0) in a simply connected region (region without holes), then the system dW/dz = F(W, P) , dP/dz = G(W, P) has no periodic solutions in this region.

Phase diagram

1. draw axis

2. draw nullclines

3. identify steady states = intersection of x/y nullclines

4. nullclines separate phase space into distinct regions, indicate signs of the rate of change

5. sketch a few example trajectories

Bifurcation diagrams

1. do model analysis

2. choose bifurcation parameter

3. draw axis with bifurcation parameter (horizontal) and solution measure (vertical)

4. Draw stable steady states

5. (optional) draw unstable steady states

6. add information on other asymptotic states (e.g. max/min)