derivation of the model it can be instructive to show how an equation like ( arises consider some population of (sayan animal species and let tbe the number of individuals in certain spatial regione an island we are not concerned with the spatial distribution of the animalsjust the number of them in some spatial area where there is no exchange of individuals with other spatial areas during time interval tsome animals will die and some new will be born the number of deaths and births are expected to be proportional to for exampleif there are twice as many individualswe expect them to get twice as many newborns in time interval tthe net growth of the population will be td bn dn / where bn tis the number of newborns and dn tis the number of deaths if we double twe expect the proportionality constants bn and dn to double tooso it makes sense to think of bn and dn as proportional to and "factor outt that iswe introduce = and dn = to be proportionality constants for newborns and deaths independent of alsowe introduce which is the net rate of growth of the population per time unit our model then becomes td rn ( equation ( is actually computational model given /we can advance the population size by td tc rn tthis is called difference equation if we know tfor some te we can compute td rn td tc rn / td tc rn /: /td ktc rn kt/where is some arbitrary integer computer program can easily compute /tfor us with the aid of little loop warning observe that the computational formula cannot be started unless we have an initial conditionthe solution of rn is rt for any constant and the initial condition is needed to fix so the solution becomes unique howeverfrom mathematical point of viewknowing tat any point is sufficient as initial