Math Bio Summary Notes
A model: What? Describes a system/process, is simplified, looks at predictions. Why? For
understanding, simplification, prediction, to see the effects of changes. How? Step by step, basic
assumptions, model parameters, simplified to only key details. What? (biological) Population, disease,
predator-prey model, enzyme substrates, evolutionary dynamics etc
Malthusian Model: dN/dt = bN – dN = rN = f(N), b=birth rate, d=death rate, r=net growth rate
Since F(0) = 0, dN/dt = NF(N), net per capital growth rate
Logistic Growth: dN/dt = rN (1 – N/k ), r=max per capita growth rate, k=carrying capacity, r, k > 0
__________
saturation: dN/dt ~ 0 Equilibria: dN/dt = rN(1-N/k) = f(N), f(N) = 0, if N=0
trivial/extinction, if N=k carrying capacity
Stability: df/dN = r ( 1 – 2N/k ), at N=0, = r > 0, at N=k,
exponential growth: dN/dt ~ rN = -r < 0
IN GENERAL: dN/dt = f(N), find equilibria at f(N*) = 0 (must be biologically realistic), stability depends on
lambda = df/dN|N=N* where if lambda < 0 it is stable, and if lambda > 0 it is unstable.
Allee Effects: describes the phenomenon of reduced per capita popn growth at low popn densities, one
of the most general models is dN/dt = rN ( 1 – N/K ) ( N/K – u/K ), u is the allee threshold – K < u < K,
weak if -K < u < 0, strong if 0 < u < K
Compensation (logistic growth): net per-capita growth rate F(N) satisfies dF/dN < 0 for all N. No allee
effects
Depensation: F(N) > 0 and an increasing function of N for small N. df/dN > 0 and F(N) > 0 for small N.
Weak allee effect
Critical depensation: F(N) < 0 and dF/dN > 0 for small N. Strong allee effect
Harvesting
Goals: balanced yield, maintain the population, exploit natural resource
Assume ‘catch’ is proportional to effort, E. Model: dN/dt = g(N) – h(N). E.g.: g(N) = rN (1-N/K), h(N) =
qEN
Bifurcation Diagram: Transcritical bifurcation, notes pg. 8
Maximum Sustainable Yield: Y = harvest rate at dynamic equil, Y = h(N*1) = qEN*1 = qEK ( 1 – qE/r ), for
fixed q, Yopt = rK/4 (at Eopt = r/2q )
Allee effect diagram and bifurcation diagram in week 2 notes
Bifurcation: N*0 = 0, always exists, L.A.S; N*1 exists if E < Ec, unstable; N*2 exists if E < Ec L.A.S; ‘saddle
node bifurcation’
Optimal might be right next to the level needed for collapse
Harvesting can shift depensation (weak allee effect) to critical depensation (a strong allee effect) [Ex. 3.1]
Hysteresis
current state of the system depends on its history
Spruce Budworm: du/dt = f(u) = ru ( 1 – u/Q ) – u2/1+u2 [1st term: g(u) log. growth, 2nd: p(u) predation]
Full analysis 2.5.1: gradually looking at increases in Q
A model: What? Describes a system/process, is simplified, looks at predictions. Why? For
understanding, simplification, prediction, to see the effects of changes. How? Step by step, basic
assumptions, model parameters, simplified to only key details. What? (biological) Population, disease,
predator-prey model, enzyme substrates, evolutionary dynamics etc
Malthusian Model: dN/dt = bN – dN = rN = f(N), b=birth rate, d=death rate, r=net growth rate
Since F(0) = 0, dN/dt = NF(N), net per capital growth rate
Logistic Growth: dN/dt = rN (1 – N/k ), r=max per capita growth rate, k=carrying capacity, r, k > 0
__________
saturation: dN/dt ~ 0 Equilibria: dN/dt = rN(1-N/k) = f(N), f(N) = 0, if N=0
trivial/extinction, if N=k carrying capacity
Stability: df/dN = r ( 1 – 2N/k ), at N=0, = r > 0, at N=k,
exponential growth: dN/dt ~ rN = -r < 0
IN GENERAL: dN/dt = f(N), find equilibria at f(N*) = 0 (must be biologically realistic), stability depends on
lambda = df/dN|N=N* where if lambda < 0 it is stable, and if lambda > 0 it is unstable.
Allee Effects: describes the phenomenon of reduced per capita popn growth at low popn densities, one
of the most general models is dN/dt = rN ( 1 – N/K ) ( N/K – u/K ), u is the allee threshold – K < u < K,
weak if -K < u < 0, strong if 0 < u < K
Compensation (logistic growth): net per-capita growth rate F(N) satisfies dF/dN < 0 for all N. No allee
effects
Depensation: F(N) > 0 and an increasing function of N for small N. df/dN > 0 and F(N) > 0 for small N.
Weak allee effect
Critical depensation: F(N) < 0 and dF/dN > 0 for small N. Strong allee effect
Harvesting
Goals: balanced yield, maintain the population, exploit natural resource
Assume ‘catch’ is proportional to effort, E. Model: dN/dt = g(N) – h(N). E.g.: g(N) = rN (1-N/K), h(N) =
qEN
Bifurcation Diagram: Transcritical bifurcation, notes pg. 8
Maximum Sustainable Yield: Y = harvest rate at dynamic equil, Y = h(N*1) = qEN*1 = qEK ( 1 – qE/r ), for
fixed q, Yopt = rK/4 (at Eopt = r/2q )
Allee effect diagram and bifurcation diagram in week 2 notes
Bifurcation: N*0 = 0, always exists, L.A.S; N*1 exists if E < Ec, unstable; N*2 exists if E < Ec L.A.S; ‘saddle
node bifurcation’
Optimal might be right next to the level needed for collapse
Harvesting can shift depensation (weak allee effect) to critical depensation (a strong allee effect) [Ex. 3.1]
Hysteresis
current state of the system depends on its history
Spruce Budworm: du/dt = f(u) = ru ( 1 – u/Q ) – u2/1+u2 [1st term: g(u) log. growth, 2nd: p(u) predation]
Full analysis 2.5.1: gradually looking at increases in Q
