Mathematical Modeling in Population Dynamics Glenn Ledder University of Nebraska-Lincoln Supported by NSF grant DUE

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

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

Mathematical Modeling Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation 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!

Example: Mars Rover Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation Conceptual Model: Newtonian physics Validation by many experiments Result: Safe landing

Example: Financial Markets Real World Conceptual Model approximationderivation analysisvalidation Conceptual Model: Financial and credit markets are independent Financial institutions are all independent Analysis: Isolated failures and acceptable risk Validation?? Result: Oops!! Mathematical Model

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 Uses data from most polls Corrects for prior pollster results Corrects for errors in pollster conceptual models Validation? Most states within 2%!

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

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.

Lotka-Volterra model x = prey, y = predator x ′ = r x – s x y y′ = c s x y – m y

Lotka-Volterra dynamics x = prey, y = predator x ′ = r x – s x y y′ = c s x y – m y Predicts oscillations of varying amplitude Predicts impossibility of predator extinction.

Logistic Growth –Fixed environment capacity K r Relative growth rate

Logistic model x = prey, y = predator x ′ = r x ( 1 – — ) – s x y y′ = c s x y – m y xKxK

Logistic dynamics x = prey, y = predator x ′ = r x ( 1 – — ) – s x y y′ = c s x y – m y Predicts y → 0 if m too large xKxK

Logistic dynamics x = prey, y = predator x ′ = r x ( 1 – — ) – s x y y′ = c s x y – m y Predicts stable x y equilibrium if m is small enough xKxK OK, but real systems sometimes oscillate.

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.

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

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 = f s x and f + h G = 1 G = —–––– = —––– s x 1 + sh x q x a + x Holling Type 2 consumption –Saturation

Holling Type 2 model x = prey, y = predator x ′ = r x ( 1 – — ) – —––– y′ = —––– – m y xKxK qx y a + x c q x y a + x

Holling Type 2 dynamics x = prey, y = predator x ′ = r x ( 1 – — ) – —––– y′ = —––– – m y Predicts stable x y equilibrium if m is small enough and stable limit cycle if m is even smaller. xKxK qx y a + x c q x y a + x

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 :

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: 1.Birds eat only infected fish. 2.Botulism death is proportional to bird population.

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

CB dynamics 1.Mutual survival possible. 2.y →0 if m+p too big. 3.Limit cycles if m+p too small. 4.I →0 if β too small. S ′ = rS ( 1− —— ) − βSI I ′ = βSI − —— − μI y ′ = —— − my − py S + I K qIy a + I cqIy a + I

CB dynamics 1.Mutual survival possible. 2.y →0 if m+e too big. 3.Limit cycles if m+e too small. 4.I →0 if β too small. BUT 5.The model does not allow the predator to survive without the disease! DUH! The birds have to eat healthy fish too!

REU 2002 corrections Flake, Hoang, Perrigo, Rose-Hulman Undergraduate Math Journal Vol 4, Issue 1, The predator should be able to eat healthy fish if there aren’t enough sick fish. 2.Predator death should be proportional to consumption of sick fish.

CB model S ′ = rS ( 1− —— ) − βSI I ′ = βSI − —— − μI y ′ = —— − my − py S + I K qIy a + I cqIy a + I Changes needed: 1.Fix predator death rate. 2.Add predation of healthy fish. 3.Change denominator of predation term.

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

FHP dynamics p > cp < cp > cp < c

FHP dynamics

p > cp < cp > cp < c