1Physiology 472/572 - 2011 - Quantitative modeling of biological systems Lecture 22: Population dynamics Discrete exponential growth • consider a batch of bacteria, whose population increases by a factor r every hour • if Pn is the population after n hours, then Pn+1 = Pnr so Pn = P0rn where r is the growth rate • example: Pn = 2n Age-structured discrete growth (Fibonacci's rabbits) • suppose each pair of rabbits born in a given season produces one pair of rabbits in each of the next two seasons only: how many pairs are born in each season? • Pn = Pn-1 + Pn-2 so {P0, P1, P0, P3,…} = {1,1,2,3,5,8,13,…} • try P = P0rn then r2 = r + 1 • solutions are r1 = (1 + √5)/2 = 1.618 (the golden ratio) and r2 = (1 - √5)/2 = -0.618 • in fact, Pn = a1r1n + a2r2n where a1 and a2 are constants Continuous exponential growth (Malthus, 1798) • consider short time intervals Δt, then the population changes only slightly over that interval, so amount of increase is proportional to Δt, i.e., r = 1 + bΔt • then P(t + Δt) = (1 + bΔt)P(t), tP(t)-t)P(tlimdtdP0tΔΔ+=→Δ, i.e., bPdtdP= • solution P = P0ebt, where b is the continuous or intrinsic growth rate • relation with discrete exponential growth: r = eb, r ≈ 1 + b if b is small • doubling time: if 2 = ebt = rt then t = ln2/b = ln2/ln(r)2r = 1.5,2,2.5,3,3.5,400.20.40.60.811 4 7 10131619222528nxnr = 2.800.5100.51xnxn+1Continuous logistic growth (Verhulst, 1838) • in reality, growth must be limited, for example by available resources or by intraspecies competition • consider )PPPb(1dtdPm−= , P(0) = P0 • solutionbt0m0m0e)P(PPPPP−−+= Discrete logistic growth • consider the discrete version Pn+1 = rPn(1 − Pn/Pmax) where r > 1 (otherwise dies out) • define xn = Pn/Pmax, then xn+1 = rxn(1 − xn) • look for a fixed point (steady-state) x such that )x1(xrx−=, i.e. 0)xrr1(x =+− • solutions x = 0 and x= 1 − 1/r • to analyze stability, consider a value close to the equilibrium • let xn = Δn, a small number, then Δn+1 ≈ rΔn so solution grows: unstable • let xn = 1 − 1/r + Δn, then xn+1 = r(1 − 1/r + Δn)(1/r − Δn) ≈ (1 − 1/r) + (2 − r)Δn i.e., Δn+1 = (2 − r)Δn, so solution is unstable if |2 - r| > 1, which occurs if r >3 • as r increases, the solutions show increasingly complex oscillations 00.20.40.60.810 0.5 1 1.5 2TimePopulation3• for r > 3.57 chaotic behavior occurs, in which a small change in initial conditions causes a large change in the solution • this is not a realistic representation of a biological system, but some more realistic systems also exhibit
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