Chapter 6 Models for Population Population models for single species –Malthusian growth model –The logistic model –The logistic model with harvest –Insect outbreak model Models for interacting populations –Predator-prey models: Lotka-Volterra systems –Competition models Other models –With age distribution –Delay models squirrels Oak trees

References J.D. Murray, Mathematical Biology, second edition, Springer-Verlag, F.C. Hoppensteadt & C.S. Peskin, Mathematics in Medicine and the Life Sciences, Springer- Verlag, 1997 A.C. Fowler, Mathematical Models in the Applied Sciences, Cambridge University Press, 1997.

Population of interaction species Three main types of interaction: –Predator-prey: growth rate of one decreased & the other increased –Competition: growth rate of both decreased –Mutualism or symbiosis: growth rate of both enhanced Predator-Prey models: –Lotka-Volterra systems: Lotka, 1925 & Volterra, 1926 –Competition models –Mutualism or Symbiois –General Models

Lotka-Volterra system Assumption: explain the oscillatory levels of certain fish catches in Adriatic –Prey in absence of any predation grows in Malthusian way –Predation is to reduce the prey’s per capita growth by a term preoperational to the prey and predator populations –In the absence of prey, the predator’s death rate is constant –The prey’s contribution to the predator’s growth is proportional to the prey & the size of the predator population t: time N(t): prey population P(t): predator population

Lotka-Volterra system Non-dimensionalization Dimensionless system Equilibrium –u=v=0 –u=v=1

Lotka-Volterra system In u, v phase plane: Phase trajectories:

Lotka-Volterra system Explanation: –A close trajectory in u,v plane implies periodic solution of u&v –The constant H determined by u(0) & v(0) –u has a turning point when v=1 & v has one when u=1

Lotka-Volterra system Trajectory plot: Lotka-Volterra Tool Different examples –Case 1: a=1, b=1, d=1, c=0 –Case 2: a=1, b=1, d=1, c=0.05 –Case 3: a=1, b=1, d=1, c=0.5 –Case 4: a=1, b=1, d=1, c=1 –Case 5: a=1, b=1, d=1, c=10

Lotka-Volterra system Jacobian matrix of the system Stability: –undetermined: u=v=1 –Unstable: u=v=0 Unrealistic: The solutions are not structurally stable!! Suppose u(0) & v(0) are such that u & v are on trajectory H4. Any small perturbation will move the solution onto another trajectory which does not lie everywhere close to H4

Unstable steady state

Lotka-Volterra system Lotka-Volterra system: –Show that predator-prey interactions result oscillatory behaviors –Unrealistic assumption: prey growth is unbounded in the absence of predation Realistic predator-prey model

Lotka-Volterra system Realistic Lotka-Volterra system: Dimensionless variables Dimensionless form

Lotka-Volterra systme Steady state populations: –u*=0, v*=0 –u*=1, v*=0 –Positive steady state: Stability of the positive steady state

Lotka-Volterra system Linear stability condition

Competition models Assumption: two species compete for the same limited food source The Model: Nondimensionalization ? Steady state ? Stability ?

Mutualism or Symbiosis Assumption: The interaction is to the advantage of all, e.g. plant or seed dispersers Nondimensionalization ? Steady state ? Stability ?

General Models Kolmogorov equations Example of three species: Lorenz (1963) –Steady state ? –Stability ? –A periodic behavior cant arise

Model with age distribution Deficiency of ODE models –No age structure & size –Birth rate & death rate depend on age! Dependence of birth rate & death rate on age

Model with age distribution Kinetic or mesoscopic model –t: time –a: age, –n(t,a): population density at time t in the age range [a,a+da] –b(a): birth rate of age a – : death rate of age a –In time range [t,t+dt], # of population of age a dies –The birth rate only contribute to n(t,0) –no births of age a>0

Model with age distribution Conservation law for the population Von Foerster equation (PDE)

Model with age distribution Characteristics: on which Integrate along the characteristic line: –When a>=t

Characteristic lines

Model with age distribution –When a<t –Where n(t-a,0) solves –It is a linear integral equation, can be solved numerically by iteration!!

Model with age distribution Similarity solution: The age distribution is simply changed by a factor ODE Plug into the boundary condition

Model with age distribution Population grow Population decay Critical threshold S for population growth S>1 implies growth & S<1 implies decay, S is determined solely b the birth & death!!

Delay models Deficiency: birth rate is considered to act instantaneously In practice: –a time delay to take account of the time to reach maturity –finite gestation period Delay Model in general Logistic delay model

Delay Models Oscillatory behaviors, e.g. Nondimensional form Steady state: N=1 Linearize around N=1

Delay models Look for solutions Unstable of N=1 since Application in physiology: dynamic diseases