Stano Pekár"Populační ekologie živočichů" dN = Nr dt continuous model of Lotka & Volterra (1925-1928) H .. density of prey P .. density of predators r .. intrinsic rate of prey population m .. predator mortality rate a .. predation rate b .. reproduction rate of predators in the absence of predator, prey grows exponentially in the absence of prey, predator dies exponentially predation rate is linear function of the number of prey .. aHP each prey contributes identically to the growth of predator .. bHP rH dt dH = mP dt dP -= aHPrH dt dH -= mPbHP dt dP -= prey population would grow to infinity neutral stability do not converge, has no asymptotic stability (trajectories are closed lines) unstable system, amplitude of the cycles is determined by initial numbers POOR model Zero isoclines: for prey population: for predator population: H P prey isocline predator isocline 0 0= dt dP aHPrH -=0 a r P = mPbHP -=0 b m H = 0= dt dH timedensity prey predator 0 Analysis of the model a r b m in the absence of the predator prey population reaches carrying capacity K Incorporation of density-dependence for given parameter values: r = 3, m = 2, a = 0.1, b = 0.3, K = 10 HP H H dt dH 1.0 10 13 - -= PHP dt dP 23.0 -= aHP K H rH dt dH - -= 1 mPbHP dt dP -= Zero isoclines: for prey population: if H = 0 (trivial solution) or if for predator population: 0.3HP - 2P = 0 if P = 0 (trivial solution) or if 0.3H - 2 = 0 if gradient of prey isocline is negative .. approached stable equilibrium H P 30 6.70 0= dt dH HP H H 1.0 10 130 - -= 0= dt dP P = 30 - 3HP H 1.0 10 130 - -= H = 6.667 10 functional response Type II: rate of consumption by all predators: Incorporation of functional response for parameters: rH = 3, a = 0.1, Th = 2, K = 10 prey isocline: h a aHT aHT H + = 1 h a aHT aHP T PH + = 1 h H aHT aHP K H Hr dt dH + - -= 1 1 0= dt dH 21.01 1.0 10 130 H HPH H + - -= 2 6.0630 HHP -+= mPbHP dt dP -= .. damped oscillations predator exploits prey close to K - isocline: H = 9 time density time density time density predator exploits prey close to K/2 - isocline: H = 5 predator exploits prey at low density - isocline: H = 2 Rosenzweig & MacArthur (1963) H P H P H P K prey predator 0 0 0 0 0 0K/2 K Damped oscillations Sustained oscillations Extinction K K logistic model with carrying capacity proportional to H k .. carrying capacity of the predator rP = bH - m Incorporation of predator's carrying capacity H P K for parameters: rP = 2, k = 0.2 predator isocline: prey isocline: mPbHP dt dP -= -= kH P Pr dt dP P 1 0= dt dP -= H P P 2.0 120 0 H = 5P 2 6.0630 HHP -+= h H aHT aHP K H Hr dt dH + - -= 1 1 discrete model of Nicholson & Bailey (1935) Ht = number of hosts in time t Ha = number of attacked hosts = finite rate of increase of the host Pt = number of parasitoids c = conversion rate, no. of parasitoids for 1 host (=1) )(1 att HHH -=+ aat HcHP ==+1 parasitoid searches randomly, has unlimited ability to lay eggs encounters (x) are random (Poisson distribution) p0 = proportion of not encountered, .. mean number of encounters Et = total number of encounters a = searching efficiency (proportion of hosts encountered) Et = a Ht Pt proportion of encounters (1 or more times): p = (1- p0) Incorporation of random search x = 0, 1, 2, ... !x e p x x - = = ep0 ( )taP ta eHH -= 1 t t t aP H E == taP ep - =0 )1( taP ep - -= highly unstable model for all parameter values: - equilibrium is possible but the slightest disturbance leads to divergent oscillations (extinction of parasitoid) taP tt eHH + = 1 ( )taP tt eHP + -= 11 time density H P 0 0 )(1 att HHH -=+ at HP =+1 exponential growth of hosts is replaced by logistic equation H*.. new host carrying capacity depends on parasitoids' efficiency - when a is low then q 1 - when a is high then q 0 density-dependence have stabilising effect for moderate r and q Stability boundaries Incorporation of density-dependence Beddington et al. (1975) t t aP K H tt eHH - + = 1 1 ( )taP tt eHP + -= 11 K H q * = Incorporation of the refuge if hosts are distributed non-randomly in the space Fixed number refuge: H0 hosts are always protected have strong stabilising effect even for large r Hassell & May (1973) taP tt eHHHH + -+= )( 001 ( )taP tt eHHP + --= 1)( 01 distribution of encounters is not random but aggregated (negative binomial distribution) - proportion of hosts not encountered (p0): where k = degree of aggregation very stable model system if k 1 Stability boundaries: a) k=, b) k=2, c) k=1, d) k=0 Incorporation of aggregated distribution Hassell (1978) k tt k aP K H tt eHH - + + = 11 1 +-= - + k t tt k aP HP 111 k t k aP p - += 10 You want to control population of mites. Before introduction of predatory mites you want to simulate the predator-prey dynamic using the following model: To estimate parameters you need to run the following experiments: 1. Keep prey population without predators, record densities over a month and estimate intrinsic rate of increase (rH) and carrying capacity (K). You found that rH = 0.2 and K = 500. 2. Keep predators at constant density of prey, record predator densities over a month and estimate natural predators' mortality (d). You found that d = 0.1. 3. Offer one predator different prey densities and estimate the functional response. You find that a = 0.001 and Th = 0.5. 4. How long it takes for the predatory mite to control mite pests if pests has a density of 200 individuals and the predators is only 1? h H aHT aHP K H Hr dt dH + - -= 1 1 dP aHT aHP dt dP h - + = 1 predprey<-function(t,y,pa){ H<-y[1] P<-y[2] with(as.list(pa),{ dH.dt<-rH*H*(1-H/K)-a*H*P/(1+a*H*Th) dP.dt<- a*H*P/(1+a*H*Th)-d*P return(list(c(dH.dt,dP.dt)))})} H<-200; P<-1 time<-seq(0,500,0.1) pa<-c(rH=0.2,K=500,a=0.001,Th=0.5,d=0.1) library(deSolve) out<-data.frame(ode(c(H,P),time,predprey,pa)) matplot(time,out[,-1],type="l",lty=1:2,col=1) legend("right",c("H","P"),lty=1:2) Aphids has increased their population density to 50 individuals/plant. You have observed that their = 3, K = 800, Th = 0.3 You need to control aphids using a parasitoid. You can choose from three parasitoid species (A, B, C). The three species differ in the number of hosts they infect (c) and in their search efficiency (a): Use the discrete Nicholson-Bailey host-parasitoid model with functional response of the type II within POPULUS. Introduce a single parasitoid and find which of the three species will achieve the quickest control. A B C c 1 2 5 a 0.3 0.07 0.001