Mathematical Modeling in Population Dynamics

1. Mathematical Modeling in Population Dynamics Glenn Ledder University of Nebraska-Lincoln http://www.math.unl.edu/~gledder1 gledder@math.unl.edu Supported by NSF grant DUE 0536508

2. Mathematical Model Math Problem Input Data Output Data Key Question: What is the relationship between input and output data?

3. Endangered Species Fixed Parameters Mathematical Model Future Population Control Parameters Model Analysis: For a given set of fixed parameters, how does the future population depend on the control parameters?

4. Mathematical Modeling Real World approximation Conceptual Model derivation Mathematical Model validation analysis • A mathematical model represents a simplified view of the real world. • We want answers for the real world. • But there is no guarantee that a model will give the right answers!

5. Example: Mars Rover Real World approximation Conceptual Model derivation Mathematical Model validation analysis • Conceptual Model: • Newtonian physics • Validation by many experiments • Result: • Safe landing

6. Example: Financial Markets Real World approximation Conceptual Model derivation Mathematical Model validation analysis • Conceptual Model: • Financial and credit markets are independent • Financial institutions are all independent • Analysis: • Isolated failures and acceptable risk • Validation?? • Result: Oops!!

7. Forecasting the Election • Polls use conceptual models • What fraction of people in each age group vote? • Are cell phone users “different” from landline users? • and so on • http://www.fivethirtyeight.com • Uses data from most polls • Corrects for prior pollster results • Corrects for errors in pollster conceptual models • Validation? • Most states within 2%!

8. General Predator-Prey Model Let x be the biomass of prey. Let y be the biomass of predators. Let F(x) be the prey growth rate. Let G(x) be the predation per predator. Note that F and G depend only on x. c, m : conversion efficiency and starvation rate

9. Simplest Predator-Prey Model Let x be the biomass of prey. Let y be the biomass of predators. Let F(x) be the prey growth rate. Let G(x) be the predation rate per predator. F(x) = rx: Growth is proportional to population size. G(x) = sx: Predation is proportional to population size.

10. Lotka-Volterra model • x = prey, y = predator • x′ = rx–sxy • y′ =csxy – my

11. Lotka-Volterra dynamics x = prey, y = predator x′ = rx–sxy y′ =csxy – my Predicts oscillations of varying amplitude Predicts impossibility of predator extinction.

12. Logistic Growth • Fixed environment capacity Relative growth rate r K

13. Logistic model x = prey, y = predator x′ = rx(1 – — )– sxy y′ = csxy – my x K

14. Logistic dynamics x = prey, y = predator x′ = rx(1 – — )– sxy y′ = csxy – my Predicts y→0 if m too large x K

15. Logistic dynamics x = prey, y = predator x′ = rx(1 – — )– sxy y′ = csxy – my Predicts stable xy equilibrium if mis small enough x K • OK, but real systems sometimes oscillate.

16. Predation with Saturation • Good modeling requires scientific insight. • Scientific insight requires observation. • Predation experiments are difficult to do in the real world. • Bugbox-predator allows us to do the experiments in a virtual world.

17. Predation with Saturation The slope decreases from a maximum at x=0 to 0 for x→∞.

18. Holling Type 2 consumption • Saturation Let s be search rate Let G(x) be predation rate per predator Let f be fraction of time spent searching Let h be the time needed to handle one prey G=fsxand f +hG =1 G =—–––– = —––– sx 1 +shx qx a+ x

19. Holling Type 2 model • x = prey, y = predator • x′ = rx(1 – — )– —––– • y′ = —––– – my x K qxy a + x cqxy a + x

20. Holling Type 2 dynamics x = prey, y = predator x′ = rx(1 – — )– —––– y′ = —––– – my Predicts stable xy equilibrium if mis small enough and stable limit cycle if miseven smaller. x K qxy a + x cqxy a + x

21. Simplest Epidemic Model Let S be the population of susceptibles. Let I be the population of infectives. Let μ be the disease mortality. Let β be the infectivity. No long-term population changes. S′ = −βSI: Infection is proportional to encounter rate. I′ = βSI−μI :

22. Salton Sea problem • Prey are fish; predators are birds. • An SI disease infects some of the fish. • Infected fish are much easier to catch than healthy fish. • Eating infected fish causes botulism poisoning. C__ and B__, Ecol Mod, 136(2001), 103: • Birds eat only infected fish. • Botulism death is proportional to bird population.

23. CB model S′=rS (1− ——) − βSI I′=βSI − —— −μI y′= —— − my− py S + I K qIy a + I cqIy a + I

24. CB dynamics S′=rS (1− ——) − βSI I′=βSI − —— −μI y′= —— − my− py S + I K • Mutual survival possible. • y→0 if m+ptoo big. • Limit cycles if m+ptoo small. • I→0 if βtoo small. qIy a + I cqIy a + I

25. CB dynamics • Mutual survival possible. • y→0 if m+etoo big. • Limit cycles if m+etoo small. • I→0 if βtoo small. • BUT • The model does not allow the predator to survive without the disease! • DUH! • The birds have to eat healthy fish too!

26. REU 2002 corrections • Flake, Hoang, Perrigo, • Rose-Hulman Undergraduate Math Journal • Vol 4, Issue 1, 2003 • The predator should be able to eat healthy fish if there aren’t enough sick fish. • Predator death should be proportional to consumption of sick fish.

27. CB model S′=rS (1− ——) − βSI I′=βSI − —— −μI y′= —— − my− py S + I K Changes needed: Fix predator death rate. Add predation of healthy fish. Change denominator of predation term. qIy a + I cqIy a + I

28. FHP model S′ = rS (1− ——) − ———— − βSI I′ = βSI − ————−μI y′ = ——————— − my S + I K qvSy a + I + vS qIy a + I + vS cqvSy + cqIy −pqIy a + I + vS Key Parameters: mortality virulence

29. FHP dynamics p > c p > c p < c p < c

30. FHP dynamics

31. FHP dynamics

32. FHP dynamics

33. FHP dynamics

34. FHP dynamics p > c p > c p < c p < c