### 4.6.1 Revised assumptions

So far, we have considered mainly population models which are continuous, that is, they use a conninuous time-variable and a continuous age-scale. The exponential and logistic models dealt with previously are examples of continuous models.

Population models also exist which use a discrete time-variable and a corresponding discrete age-scale. Naturally, since the information used in a discrete model is necessarily grouped we will need to know how to calculate both the mortality rates and fertility rates for the groups before we can use the model to predict the behaviour of an observed population. In fact, these rates tend to depend on the age distribution of the population within each group and not simply on the age distribution as a wholl. One of the best known discrete model formulations is that proposed by P.H. Leslie which was first published by him in 1945. The Leslie model predicts the age structure of a population of animals after a unit period of time has passed given that the structure at the start of the time period is know and the appropriate information regarding mortality and fertility rates is known. The initial development of the model is outlined below.

Firstly, since only the females in a population can give birth, the initial model considers the female population only. In effect this is our first assumption, you might like to comment on the validity of this assumption since it assumes that the male population has no effect whatsoever on the fertility rates of the grouped populations.

Secondly, we break the female population up into age groups corresponding to the time intervals t = 0,1,2,3,4,...... . We assume that there are m+1 age groups denoted by 0-,1-,2-,3-,...m-. The number of females in age group x at time t is denoted by nx,t.

Mortality Rates

The proportion of females surviving group x to become members of group x+1 at time t+1 is denoted by Px. Notice that whilst Pm = 0 , since the proportion surviving the last age group must be zero by definition.

Fertility Rates

The average number of female fishes born to the members of group x at time t is denoted by Fx. We assume that all of these survive to become members of group 0- at time t+1. Essentially, we are assuming that the mortality rate of is zero.

Generally, we assume that the mortality rates and fertility rates stay constant over several time periods, not just one.