ing stage (larvae and adults) move through the flour and may encounter individuals in a non-moving stage, such as eggs and pupae. When this happens, the moving larva or adult beetle will bite the egg or pupa, thereby killing it. We assume that larvae only kill eggs, since pupae are biggerthan larvae. Suppose the probability that an adult encoun ters one egg during the period of time At is given by pAt, where p is constant. Then, the probability for this egg not to have been eaten by this adult at the end of At is 1 pAt, and the probability for this (or any other) egg to become a larva is then (1 pAt)A(t), since there are A(t) adults. Thus, if eggs were only eaten by adults, one would write L(t + At) = bA(t)(1 pAtW = Wt) \"PW\" 1\"\" ' 1' A\") = bA(t) exp(CeaA(t)), where cw = 111(1 pAt). Following similar arguments, the fact that eggs are also eaten by larvae is taken into account by multiplying the right hand side of the above equation by exp(celL(t)). Therefore, the deterministic LPA model (where LPA stands for Larva, Pupa, Adult), which includes cannibalism of pupae and eggs by adults and of eggs by larvae, reads L(t + 1) : bA(t) exp(_ceaA(t) _ celL(t))7 P(t + 1) = L(t) dlL(t), A(t -l 1) = A(t) dA(t) + P(t) exp(cpaA(t)). A review of the dynamical properties of the LPA model can be found in the 2004 article by Jim Cushing et al. entitled Nonlinear population dynamics: mode/s, experiments and M. In particular, it can be shown that the trajectories of the LPA model which start in the non-negative octant remain non-negative. Moreover, when stochastic terms are added to the above equations, the predictions given by the resulting model, including behavior consistent with chaotic dynamics, are in particular good agreement with experiments. The LPA Model Depending on the type of species whose populations is being modeled, effects other than births and deaths may have to be considered. As an illustration, we now discuss a 91515] for a population of model developed by RF. Costantino and colleaguesl flour beetles, for which adults are known to eat their offspring. The different stages in the development of a beetle are the larval, pupal and adult stages. If one chooses At (about 2 weeks for flour beetles) such that it corresponds to the time it takes on aver age for a larva to pupate and for a pupa to become a reproductive adult, one has, counting time in units of A15, and in the absence of cannibalism, L(t + 1) = bA(t), P(t + 1) = L(t) _ d;L(t), A(t + 1) = Am dam) + Pm, where the positive constants d; and d.\" are the probabilities of death for larvae and adults during the period At (= 1), the parameter b > 0 is the average number of eggs per adult that have hatched during At, and one has neglected deaths in the pu pal stage. Cannibalism of eggs by adults and larvae is modeled through multiplicative terms of the form exp(cmA(t)) and exp(celL(t)) respectively, and cannibalism of pupae by adults is described by exp(cmA(t)) (see the articles by Costantino et al. and Cushing et al. mentioned above). Here, the constants cm, cal and cm are all positive. The exponential terms represent the fraction of eggs or pupae which survive to the next stage in their development, and can be understood as follows. Beetles in a crawl- ing stage (larvae and adults) move through the flour and may encounter individuals in a nonmoving stage, such as eggs and pupae. When this happens, the moving larva or adult beetle will bite the egg or pupa, thereby killing it. We assume that larvae only kill eggs, since pupae are bigger than larvae. Suppose the probability that an adult encoun- ters one egg during the period of time At is given by pAt, wherep is constant. Then, Problem 11 Consider the LPA model described in this chapter. What are the dimensions of the various parameters (b, Cea, Cel, Cpa, da, di) appearing in this model