Lecture 2 Modelling the dynamics of populations. First order growth and decay. The logistics equation.

BEST – IST, 2006 What is a model? Mathematical models improve with age.....

Princípio de conservação The rate of accumulation inside a control volume = What flows in minus what flows out + Production minus destruction

Closed (isolated) Volume If β is uniform inside the volume: Writting the source/sink terms per unit of volume one gets:

Population Dynamics ( n=0) => zero orde decay/growth (linear evolution ) (n=1) => 1st order (exponential) …….. c0 c t K>0 K<0 If (n=1) => 1st order : K >0 implies exponential growth K<0 asymptotic decay towards zero. In case (n=1) => 1st order : The analytical solution is:

1 st order decay Usually it is assumed that: – faecal bacterian mortality is a 1 st order decay accounted using the T90, – Pesticides have a 1 st order decay accounted by the half-life time. How to compute k? => T90=1 hour=> k=-6.4E-4 s -1. In case of the half-life time the calculation is identical, using ln(0.5)

The “Logistic”solution Cmax C0 c t The solution so called "Logistic “ solution admits that the exponential growth is not sustainable. Admitting that there is a maximum population and consequently K must be variable.

Numerical solution(explicit) Using the explicit method: Discretizing the time derivative one gets:

Comparison of the numerical and analytical solutions See the Excel workbook “dinâmica de populações”

Numerical solution (explicit) In the explicit solution we got: If k<o then the parentheses can be negative if the time step is high. In this case the new concentration would be negative and the method would be unstable. The stability condition is: In this passage the inequality sign change when we divide by k<0

Numerical Solution (implicit) In this case one gets: Now instabillities can appear if k>0:

Criteria for Stability When we have mortality, if the method is explicit, the number of individuals who dies is a function of the value that we had at the beginning of the time step. This implies that the mortality value is overestimated. If the step time is too large we can eliminate more individuals than existing ones and we get a negative value (the same can be said for the concentration). When we have growth problem arises in the implicit method because physically the number of children is in proportion to the number of parents and the calculation must be explicit. The implicit calculation would be equivalent to saying that "the kids would be born bringing children on their lap".

Generalizing one can say that: The sources must be calculated explicitly and sinks must be calculated implicitly for model stability. If the model is stable which time step should be used? A time step small enough to assure that the numerical solution does not move away from the analytical solution. c0 c t K>0 K<0 implícito explícito

Final Considerations The models based on first order decays can be realistic for properties that do not arise in the natural environment and consequently when they are released they decay continuously. The models based on first-order growth are unrealistic. The logistic equation can give them some realism. Models must reproduce the processes of production and decay. The Lotka-Volterra model is the simplest that attempts to achieve this goal.

Prey-Predator Model (Lotka-Volterra) In the equation: only logistics could limit grow. In the real world there is always predator that also contributes to limit the prey. Lotka-Volterra Equations

Lotka Volterra Model Limitations It does not conserve the total mass. A very simple nature would require at least 3 state variables: Note: The derivatives are now total derivatives to describe the case of a material system in motion. Could k p beconstant? Is it reasobable that the prey is fed with detritus? We need extra variables...

Generic shape of the evolution equations In these equations have added the difusive transport.