Stochastic Population Systems

Xuerong FRSE Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH

Outline Stochastic Modelling 1 Stochastic Modelling 2 Linear Model Nonlinear Models 3

One of the important problems in life science is the specification of the stochastic process governing the behaviour of an underlying quantity. We here use the term underlying quantity to describe any interested object whose value is known at present but is liable to change in the future. Typical examples are number of cancer cells, number of HIV infected individuals, number of people who have the swine flu.

Now suppose that at time t the underlying quantity is x(t). Let us consider a small subsequent time interval dt, during which x(t) changes to x(t) + dx(t). (We use the notation d for the small change in any quantity over this time interval when we intend to consider it as an infinitesimal change.) By definition, the intrinsic growth rate at t is dx(t)/x(t). How might we model this rate?

If, given x(t) at time t, the rate of change is deterministic, say R = R(x(t), t), then dx(t) x(t) = R(x(t), t)dt. This gives the ordinary differential equation (ODE) dx(t) dt = R(x(t), t)x(t).

However the rate of change is in general not deterministic as it is often subjective to many factors and uncertainties e.g. system uncertainty, environmental disturbances. To model the uncertainty, we may decompose dx(t) x(t) = deterministic change + random change.

The deterministic change may be modeled by Rdt = R(x(t), t)dt where R = r(x(t), t) is the average rate of change given x(t) at time t. So dx(t) x(t) = R(x(t), t)dt + random change. How may we model the random change?

In general, the random change is affected by many factors independently. By the well-known central limit theorem this change can be represented by a normal distribution with mean zero and and variance V 2 dt, namely random change = N(0, V 2 dt) = V N(0, dt), where V = V (x(t), t) is the standard deviation of the rate of change given x(t) at time t, and N(0, dt) is a normal distribution with mean zero and and variance dt. Hence dx(t) x(t) = R(x(t), t)dt + V (x(t), t)n(0, dt).

A convenient way to model N(0, dt) as a process is to use the Brownian motion B(t) (t 0) which has the following properties: B(0) = 0, db(t) = B(t + dt) B(t) is independent of B(t), db(t) follows N(0, dt).

The stochastic model can therefore be written as or dx(t) x(t) = R(x(t), t)dt + V (x(t), t)db(t), dx(t) = R(x(t), t)x(t)dt + V (x(t), t)x(t)db(t) which is a stochastic differential equation (SDE).

Outline Stochastic Modelling Linear Model Nonlinear Models 1 Stochastic Modelling 2 Linear Model Nonlinear Models 3

Linear model Stochastic Modelling Linear Model Nonlinear Models If both R and V are constants, say R(x(t), t) = µ, V (x(t), t) = σ, then the SDE becomes dx(t) = µx(t)dt + σx(t)db(t). This is the exponential growth model in population.

Outline Stochastic Modelling Linear Model Nonlinear Models 1 Stochastic Modelling 2 Linear Model Nonlinear Models 3

Logistic model Stochastic Modelling Linear Model Nonlinear Models If R(x(t), t) = b + ax(t), V (x(t), t) = σx(t), then the SDE becomes ( ) dx(t) = x(t) [b + ax(t)]dt + σx(t)db(t). This is the well-known Logistic model in population.

Square root process Linear Model Nonlinear Models If R(x(t), t) = µ, V (x(t), t) = σ x(t), then the SDE becomes the well-known square root process dx(t) = µx(t)dt + σ x(t)db(t).

Mean-reverting model Linear Model Nonlinear Models If R(x(t), t) = then the SDE becomes α(µ x(t)), V (x(t), t) = σ, x(t) x(t) dx(t) = α(µ x(t))dt + σ x(t)db(t). This is the mean-reverting square root process in population.

Theta process Stochastic Modelling Linear Model Nonlinear Models If R(x(t), t) = µ, V (x(t), t) = σ(x(t)) θ 1, then the SDE becomes dx(t) = µx(t)dt + σ(x(t)) θ db(t), which is known as the theta process.

In the classical theory of population dynamics, it is assumed that the grow rate is constant µ. Thus dx(t) x(t) = µdt, which is often written as the familiar ordinary differential equation (ODE) dx(t) = µx(t). dt This linear ODE can be solved exactly to give exponential growth (or decay) in the population, i.e. x(t) = x(0)e µt, where x(0) is the initial population at time t = 0.

We observe: If µ > 0, x(t) exponentially, i.e. the population will grow exponentially fast. If µ < 0, x(t) 0 exponentially, that is the population will become extinct. If µ = 0, x(t) = x(0) for all t, namely the population is stationary.

However, if we take the uncertainty into account as explained before, we may have dx(t) x(t) = rdt + σdb(t). This is often written as the linear SDE It has the explicit solution dx(t) = µx(t)dt + σx(t)db(t). x(t) = x(0) exp [ ] (µ 0.5σ 2 )t + σb(t).

Recall the properties of the Brownian motion lim sup t B(t) 2t log log t = 1 a.s. and lim inf t B(t) 2t log log t = 1 a.s.

If µ > 0.5σ 2, x(t) exponentially with probability 1, i.e. the population will grow exponentially fast. If µ < 0.5σ 2, x(t) 0 exponentially with probability 1, that is the population will become extinct. In particular, this includes the case of 0 < µ < 0.5σ 2 where the population will grow exponentially in the corresponding ODE model but it will become extinct in the SDE model. This reveals the important feature - noise may make a population extinct. If µ = 0.5σ 2, lim sup t x(t) = while lim inf t x(t) = 0 with probability 1.

The logistic model for single-species population dynamics is given by the ODE dx(t) dt = x(t)[b + ax(t)]. (3.1)

If a < 0 and b > 0, the equation has the global solution x(t) = b a + e bt (b + ax 0 )/x 0 (t 0), which is not only positive and bounded but also lim x(t) = b t a. If a > 0, whilst retaining b > 0, then the equation has only the local solution b x(t) = a + e bt (0 t < T ), (b + ax 0 )/x 0 which explodes to infinity at the finite time T = 1 ( ) b log ax0. b + ax 0

Once again, the growth rate b here is not a constant but a stochastic process. Therefore, bdt should be replaced by bdt + N(0, v 2 dt) = bdt + vn(0, dt) = bdt + vdb(t), where v 2 is the variance of the noise intensity. Hence the ODE evolves to an SDE ( ) dx(t) = x(t) [b + ax(t)]dt + vdb(t). (3.2)

The variance may or may not depend on the state x(t). We first consider the latter, say v = σx(t). Then the SDE (3.2) becomes ( ) dx(t) = x(t) [b + ax(t)]dt + σx(t)db(t). (3.3) How is this SDE different from its corresponding ODE?

Significant Difference between ODE (3.1) and SDE (3.3) ODE (3.1): The solution explodes to infinity at a finite time if a > 0 and b > 0. SDE (3.3): With probability one, the solution will no longer explode in a finite time, even in the case when a > 0 and b > 0, as long as σ 0. Moreover, the stochastic population system is persistent.

Example Stochastic Modelling dx(t) = x(t)[1 + x(t)], t 0, x(0) = x 0 > 0 dt has the solution x(t) = 1 1 + e t (1 + x 0 )/x 0 (0 t < T ), which explodes to infinity at the finite time ( ) 1 + x0 T = log. However, the SDE x 0 dx(t) = x(t)[(1 + x(t))dt + σx(t)dw(t)] will never explode as long as σ 0.

200 150 (a) 100 50 0 0 2 4 6 8 10 80 60 (b) 40 20 0 0 2 4 6 8 10

Note on the graphs: In graph (a) the solid curve shows a stochastic trajectory generated by the Euler scheme for time step t = 10 7 and σ = 0.25 for a one-dimensional system (3.3) with a = b = 1. The corresponding deterministic trajectory is shown by the dot-dashed curve. In Graph (b) σ = 1.0.

Key Point: When a > 0 and ε = 0 the solution explodes at the finite time t = T ; whilst conversely, no matter how small ε > 0, the solution will not explode in a finite time. In other words, noise may suppress explosion.

If the variance is independent of the state x(t), namely v = σ = const., then the SDE (3.2) becomes ( ) dx(t) = x(t) [b + ax(t)]dt + σdb(t). (3.4) How is this SDE different from others?

For this SDE, we require a < 0 in order to have no explosion. If b < 0.5σ 2 then lim t x(t) = 0 with probability 1, namely the noise makes the population extinct. If b > 0.5σ 2, the population is persistent and 1 T lim x(t)dt = b 0.5σ2 T T 0 a a.s.

Noise changes the behaviour of population systems significantly. Noise may suppress the potential population explosion. Noise may make the population extinct. Noise may make the population persistent.