Lecture 2

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

2. What is a model? Mathematicalmodels improve with age..... BEST – IST, 2006

3. Princípio de conservação The rate ofaccumulationinside a control volume Productionminusdestruction Whatflows in minuswhatflows out = +

4. Closed (isolated) Volume • Ifβisuniforminside the volume: • Writting the source/sinkterms per unitof volume onegets:

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

6. 1storderdecay • Usuallyitisassumedthat: • faecalbacterianmortalityis a 1storderdecayaccountedusing theT90, • Pesticides have a 1storderdecayaccountedby 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 calculationisidentical, usingln(0.5)

7. The “Logistic”solution • 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 bevariable. c Cmax C0 t

8. Numerical solution(explicit) Discretizing the time derivativeonegets: Using the explicitmethod:

9. Comparisonof the numericalandanalyticalsolutions See the Excel workbook “dinâmica de populações”

10. Numerical solution (explicit) In the explicitsolutionwegot: Ifk<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

11. Numerical Solution (implicit) In this case onegets: Nowinstabillities can appearif k>0:

12. 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".

13. Generalizingone can saythat: • 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. c K>0 implícito c0 explícito K<0 t

14. Final Considerations • The models based on first order decays can be realistic for properties that do not arise in the natural environmentand 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.

15. 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-VolterraEquations

16. LotkaVolterraModelLimitations • It does not conserve the total mass. A verysimplenaturewouldrequireatleast 3 statevariables: • Note: The derivatives are now total derivatives to describe the case of a material system in motion. • Couldkpbeconstant? Is itreasobablethat the preyisfedwithdetritus? Weneed extra variables...

17. Genericshapeof the evolutionequations In these equations have added the difusive transport.