if the dominant eigenvalue X of the coefficient matrix
0 1 - jXp 1 - ¡J, a is less than 1 in magnitude. (By the dominant eigenvalue A. we mean the eigenvalue of largest magnitude, which turns out to be positive by the Perron-Frobenius theorem [23,32, 68].) On the other hand, if A > 1 then (nonzero, nonnegative) solutions grow exponentially without bound. It turns out X > 1 if and only if R0 > 1 where
Ro = b lia is the net reproductive number [32,34]. It is also true that X < 1 if and only if Rq < 1.
In the case of nonextinction (as is typically observed under standard laboratory conditions), the linear Leslie model (2.1) cannot, of course, predict the long-term (asymptotic) dynamics of flour beetle populations. In order to do that, the model must include the controlling mechanism of cannibalism. Although not biologically complete, an approximation to a particular cannibalistic interaction can be described as follows. Consider the cannibalism of eggs by adults. It has been documented by laboratory experiment that the probability of a contact between an individual adult and an egg, during a fixed time interval, is inversely proportional the habitat volume V  . If we assume that during a small interval of time A t this probability is (approximately) proportional to At (and that upon such a contact there is a fixed probability that the egg is eaten), then the probability an egg will survive being eaten by one adult during At time units is approximately 1 - ^At. The probability it will survive 2 At units of time is approximately the product
and so on. The probability it will survive a full unit of time is approximately
Under the assumption that encounters between the egg and other adults are independent events, the survival probability of one egg over one unit of time in the presence of two adults is approximately
In the presence of At adults the probability that an egg survives adult cannibalism is approximately
A,/At which, as Af 0, approaches the exponential expi-^f-Â,). Thus, in our model we take the probability that an egg is not eaten in the presence of At adults to be this exponential term. We refer to the ratio ct,a/ V as the coefficient of adult cannibalism on eggs (in a habitat of volume V). In a unit of volume V the adult cannibalism coefficient is cea.
Similar exponential terms describe the probabilities of surviving cannibalism that occurs among the other life-cycle stages. The dominant cannibalistic interactions in populations of T. castaneum are egg cannibalism by larvae and adult and pupa cannibalism by adults . Introducing these survival probabilities into the linear Leslie model (2.1) we obtain the nonlinear matrix model
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