Stano Pekár"Populační ekologie živočichů" dN = Nr dt if then logistic growth due to density dependent changes in fecundity and survival K .. carrying capacity, upper limit of population growth, where = 1 change in depends on N 0 1 KNt Nt/Nt+1 1/ y = a + x b Discrete (difference) model K N N N t t t )1( 1 1 - + =+ t t t aN N N + =+ 1 1 K a 1- = tt NN =+1 1 1 = +t t N N ( ) K N N N t t t 111 1 - += + time0 K Nt when Nt 0 then * no competition * exponential growth when Nt K then * density-dependent control * S-shaped (sigmoid) growth when Nt > K then * population returns to K 1 1 < + taN 1 1 + taN + taN1 - when N K then r 0 N K dN/dt*1/N r 0 logistic growth first used by Verhulst (1838) to describe growth of human population Solution of the differential equation Continuous (differential) model -= K N Nr dt dN 1 Nr dt dN = r Ndt dN = 1 K r Nr Ndt dN -= 1 rtat e K N - + = 1 - = 0 0 ln N NK a Monotonous increase ( = 0.5) Damping oscillations ( = 1.9) Limit cycle ( = 2.3) Chaos ( = 3.0) t t t aN N N + =+ 1 1 1. N = 0 .. unstable equilibrium 2. N = K .. stable equilibrium .. if 0 < < 2 "Monotonous increase" and "Damping oscillations" has a stable equilibrium "Limit cycle" and "Chaos" has no equilibrium < 2 .. stable equilibrium = 2 .. 2-point limit cycle = 2.5 .. 4-point limit cycle = 2.692 .. chaos chaos can be produced by deterministic process density-dependence is stabilising only when is rather low Model equilibria N a) yeast (logistic curve) b) sheep (logistic curve with oscillations) c) Callosobruchus (damping oscillations) d) Parus (chaos) e) Daphnia of 28 insect species in one species chaos was identified, one other showed limit cycles, all other were in stable equilibrium Hassell (1975) proposed general model for DD Effect of on population density 0 200 400 600 800 1000 1200 1400 0 10 20 30 40 50 60 Time Density - where .. the strength of competition >> 1 .. scramble competition (over-compensation) = 1 .. contest competition (exact compensation) < 1 .. under-compensation ( ) t t t aN N N + =+ 1 1 N K r 0 = 1 = 3 = 0.2 = 0.5 = 1 = 2 species response to resource change is not immediate but delayed due to maternal effect, seasonal effect appropriate for species with long generation time where reproductive rate is dependent on density of a previous generation time lag (d, ) .. negative feedback of the 2nd order discrete model continuous model many populations of mammals cycle with 3-4 year periods time-lag provokes fluctuations of certain amplitude at certain periods period of the cycle in continuous model is always 4 dt t t aN N N - + + = 1 1 K NK rN dt dN t t -- = r < 0.4 monotonous increase r < 1.6 damping fluctuations r < 2 limit cycle fluctuations r > 2 extinction Solution of the continuous model: 0 500 1000 1500 2000 2500 0 10 20 30 40 50 Time Density tau=2 tau=6 tau=8 tau=11 K = 500 r = 0.5 - + = K N r tt t eNN 1 1 to attain maximum sustainable yield 01 = -= K N Nr dt dN 0 2 =- K rN Nr 0 2 =- K rN r 2 * rK N = 2 * K N = time Nt 0 K K/2 N* K2 .. extinction threshold, unstable equilibrium population increase is slow at low density but fast at high density - -= 11 21 K N K N Nr dt dN population sizeK2 K1 r birth death 0 Simulate population dynamics using density-dependent model for discrete population growth for a period of 40 generations with N0=10. 1. With deterministic and K ( = 1.2, K =500). 2. With stochastic but deterministic K ( = 1.2 0.12, K = 500). 3. With stochastic K but deterministic ( = 1.2, K = 500 50). 4. With stochastic and K ( = 1.2 0.12, K = 500 50). 5. Make a function for a density-dependent model for discrete population growth using N0=100, K=500, =1.2, t=50. 6. Plot logarithm of lambda against Nt and estimate max and K using the following formulas: ( ) tbNa -=ln a e=max b a K -= N<-10 for(t in 1:40) N[t+1]<-{ N[t]*1.2/(1+N[t]*(1.2-1)/500)} plot(0:40,N,type="b") for(t in 1:40) N[t+1]<-{ N[t]*1.2/(1+N[t]*(1.2-1)/runif(1,450,550))} plot(0:40,N,type="b") for(t in 1:40) N[t+1]<-{ N[t]* runif(1,1,1.5)/(1+N[t]*(runif(1,1,1.5)-1)/500)} plot(0:40,N,type="b") for(t in 1:40) N[t+1]<-{ N[t]* runif(1,1,1.5)/(1+N[t]*(runif(1,1,1.5)-1)/runif(1,450,550))} plot(0:40,N,type="b") logistic<-function(N0=100,K=500,R=1.2,t=50){ N<-c(N0,numeric(t)) for(t in 1:t) N[t+1]<-{ N[t]*R/(1+N[t]*(R-1)/K)} return(N)} Nt<-logistic() plot(Nt) lambda<-Nt[-1]/Nt[-51] plot(Nt[-51],log(lambda),ylim=c(0,0.5)) m1<-lm(log(lambda)~Nt[-51]) coef(m1) exp(0.1763726056) -0.1763726056/-0.0003536496 You have observed the following population dynamic of yearly censuses of aphids: 180, 531, 277, 296, 828, 329, 397, 772, 625, 318, 567, 881, 386 1. Plot the population dynamic. 2. Is there evidence for density-dependence? 3. Estimate max and K. aphid<-c(180, 531, 277, 296, 828, 329, 397, 772, 625, 318, 567, 881, 386) plot(aphid,type="b") lambda1<-aphid[-1]/aphid[-13] plot(aphid[-13], log(lambda1)) m2<-lm(log(lambda1)~aphid[-13]) coef(m2) abline(m2) exp(1.3057703) -1.30577030/-0.00248398 On an African market wild game animals are sold. You know carrying capacities (K), finite growth rates (R), and longevities (L) for each species: 1. Compute MSY for each species using the following formula: a = 0.6 for L < 5 where a = 0.4 for L = (5,10) a = 0.2 for L > 10 2. Is the observed harvest sustainable in each species? K R Longevity Harvest monkey 49000 1.17 31 781 pangolin 22000 2.01 13 192 porcupine 110000 1.82 23 1580 duiker 45000 1.63 7 732 - = 2 KRK aMSY monkey<-0.2*(1.17*49000-49000)/2; monkey pangolin<-0.2*(2.01*22000-22000)/2; pangolin porcupine<-0.2*(1.82*110000-110000)/2; porcupine duiker<-0.4*(1.63*45000-45000)/2; duiker