Transcription of Section 3.3: Nicholson-Bailey Model
1 Section : Nicholson-Bailey ModelThe Nicholson-Bailey Model is a discrete-time Model for two interacting insect species,a parasitoid and its host. The term parasitoid means a parasite which is free living as anadult but lays eggs in the insect host. Hosts that are not parasitized give rise to their ownprogeny. Hosts that are successfully parasitized die but the eggs laid by the parasitoid maysurvive to be the next generation of the population density of the host species in generationtandPtdenotethe population density of the parasitoid species in generationt. The Nicholson-Bailey modelis given byNt+1=rNte aPt,Pt+1=bNt(1 e aPt).The parameterr >0 is the number of eggs laid by a host that survive to the adult the absence of parasitoids (Pt= 0), the host population exhibits exponential growth ordecay. The terme aPtrepresents the fraction of hosts which are not parasitized, so that1 e aPtis the fraction of hosts which are parasitized.
2 The parameterb >0 is the numberof eggs laid by a parasitoid on a single host that survive to the adult :Consider the Nicholson-Bailey modelNt+1=rNte aPt,Pt+1=bNt(1 e aPt).(a) Find all biologically-relevant equilibria of the equilibrium ( N , P) = (0,0) is called theextinction equilibriumand( N , P) =(rlnr(r 1)ab,lnra)is called thecoexistence equilibrium. Note that the second equilibrium is positive ifr >1.(b) Determine conditions for the local asymptotic stability of the extinction equilibrium.(c) Determine conditions for the local asymptotic stability of the coexistence :The positive equilibrium of the Nicholson-Bailey Model , where( N , P) =(rlnr(r 1)ab,lnra)forr >1 is :The figures below illustrate the dynamics of the Nicholson-Bailey Model for 0< r <1andr >1. Ifr >1, the oscillations increase in amplitude, but they also get very close tozero so that numerically the values are set to zero. Population extinction occurs even in SizeNicholson- bailey Model , r<1 HostParasitoid0102030 Time0100200300400500600 Population SizeNicholson- bailey Model , r>1 HostParasitoid3
