The population size of a population of leopard geckos under certain circumstances can be modeled by the sequence{an}∞n=1 wherean+1 = panan + qwhere p and q are positive constants. an is the number of geckos in the population after n years. The size of thepopulation in the long term depends on the relative values of p and q, as well as on the initial population a1.(a) (3 marks) Zara writes down the following argument:Zara’s reasoning: If the sequence {an}∞n=1 converges, suppose it converges to some number L. Inthat case:limn→∞ an = L and also limn→∞ an+1 = LSince all the terms are positive, we must have L > 0. Therefore:L = limn→∞ an+1 = limn→∞panan + q = limn→∞p1 + q/an= p1 + q/LTherefore, if the sequence converges, then the limit L satisfies L = p1 + q/L .There is ONE error is Zara’s reasoning. Find the error, then solve for L to find all possible values ofL.(b) (3 marks) Suppose that p < q. Find the limit of limn→∞ an. What does this mean for the leopard geckopopulation?Hint: First show that an+1 < pq an, then use a known theorem together with your result from part (a).(c) (3 marks) Suppose now that p > q and a1 < p − q. This implies that an < p − q for all n (you do not needto show that, but it can be done for example by using induction, or in other ways).Find the limit of limn→∞ an. What does this mean for the population?Hint: Show that {an}∞n=1 is increasing and use a known theorem together with your result from part (a).

Suppose now that p > q and a1 < p − q. This implies that an < p − q for all n (you do not needto show that, but it can be done for example by using induction, or in other ways).Find the limit of limn→∞ an. What does this mean for the population?Hint: Show that {an}∞n=1 is increasing and use a known theorem together with your result from part (a).

In what situation might a continuous-time exponential growth model be a GOOD description of how population size grows over time? Group of answer choicesA population of animals infested by transmissible deadly parasitesA population of birds with several predators.A bacterial culture growing in a laboratory with an unlimited amount of resources.A population of rats living on a small island.

In what situation might a continuous-time exponential growth model be a BAD description of how population size grows over time?Group of answer choicesA bacterial culture growing in a laboratory with an unlimited amount of resources.A yeast culture being grown by a PhD student who forgot to feed it before going on holiday.The number of people infected by a virus just after the virus was introduced into the population.The initial phase of a population of rats just after a small number of rats arrived on an island where no rats previously lived.

The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth modeln(t) = 2,6000.5 + 25.5e−0.041twhere t is measured in years.(a) Find the initial bird population. birds(b) Draw a graph of the function n

Calculate the population changes for 25 generations (use Excel) in a hypothetical predator-prey system with discrete generations in which the parameters for the prey areB=0.03, equilibriumN eq ​ (or carrying capacityK)=100, and the starting density is 50 prey and for the predators,C=0.5,Q=0.02, and starting density is 0.2 . Graph the population densities of the predator and prey as a function of number of generations. Briefly explain how the interaction between the predator and prey could create the population dynamics produced by the model (only consider the dynamics between generations 5 and 25). Keeping the same parameters for the prey as above, show graphically and explain in words how the prey population changes in the absence of the predator.