To develop this model we shall consider the incremental effects upon the population sizes due to increases over a short period of time; system differ- ential equations will be derived from the elementary limiting process. Let N1 denote the number of individuals of the prey species and (Jl > 0 its coefficient of autoincrease. We shall assume that if the prey were isolated, the increase in its population would be proportional to its number and to the length of the time interval; that is,
(9)
The predacious species, on the other hand, would decrease in isolation because of a lack of food; hence, we have for (J2 >0,
(10) When these two species occupy a common territory, the number of en- counters between a predator and a prey is proportional to the product of the two populations sizes and to the time interval; that is the number of encounters is kN1N2l:i.t with k >O. A certain fraction (Yl of these encoun- ters result in the death of a prey, thereby decreasing the quantity given by Eq. (9) by (YlkNIN2I:i.t. The consumption of a prey contributes to the increase of the predators, and the quantity given by Eq. (10) increases by a factor f32(YlkNIN2I:i.t, where fh > 0 might be considered as the coeffi- cient of food utilization. Now, for the economy of notation, put (Yl k =/1
and fhalk =12; obviously 11 > 0 and 12 > O. Several properties of the coefficients should be noted. If the offensive mechanism of the predators improves, then the constant al increases, thereby causing both 11 and 12 to increase. If, in some manner, the prey are protected, then al decreases and both 11 and 12 decrease.
Incorporating the preceding terms into Eqs. (9) and (10) and taking the limit gives the differential equations for two species living in a predator-prey relationship:
dNI
& =(0"1 -/IN 2)N1 dN2
& =(-0"2 +12N l)N2
(11 ) (12) These equations could have been presented straight away, but Volterra's
"method of encounters" used here offers insight into the meaning of the coefficients11and12,which are composed of more fundamental parameters.
A thorough investigation of the behavior of Eqs. (11) and (12) is outside the scope of this article and can be found, e.g. in [1]. Here we shall state only the final conclusions which can be drawn from such an analysis in the following three laws:
Law of Periodicity. The fluctuations of the population sizes of both the predators and the prey are periodic, and the period depends only upon the coefficients of autoincrease (0"1,0"2), the coefficients of interaction (-rl,12) and an integration constant (C) dependent upon the initial population SIzes.
Law of Conservation of the Means. The mean value of the number of either species over anyone complete cycle is independent of the initial con- ditions and depends only upon the coefficients of increase and interaction.
The mean value of N 1is 0"21/2; the mean value ofN 2is O"tI/l'
Law of Perturbation of the Means.Ifmembers of both species are uniformly destroyed in proportion to the relative population sizes, then the mean number of the predators decreases. Both means are increased if the prey is protected in a manner which does not destroy the predators.
3.2.2. Consideration of External Forces that Influence the Predator and the Prey Populations
External forces, such as the destruction of the predators at a uniform rate, were taken into account in the previous section by varying the coefficients without too much concern for the exact nature of the variations. Also, a law concerning the perturbations of mean values by external forces was stated but was subject to the restriction0"1 >O. In that case, changes in 0"1 and 0"2 alter the mean values but do not affect the basic oscillatory nature of the
populations. In this section, we shall treat the problem of external forces in a more explicit manner, following [1].
New parameters will be defined in order to make explicit each force contributing term to the system equations. Let 0"1 > 0and 0"2 > 0 be the coefficients of autoincrease and1'1 >0 and 12 > 0 be the coefficients of in- teraction. During the time increment f:1t, let f:1N{ of the prey be destroyed by external forces; we shall assume that the number destroyed is propor- tional to the number of prey and to the length of the time increment; that is f:1N{ = a)..N1f:1t, where a > 0and)" >O. The reason for using two coef- ficients to characterize destruction of prey will be seen presently. Similarly, we have f:1N~ =(3)..N2f:1t, (3 > O. The quantity f:1NUN1 is the fraction of prey destroyed during a given time increment. The ratio of the fraction of predators destroyed by external forces to the fraction of prey destroyed is
f:1NYN2 = ~ =8
ãf:1NUNl a-
The constant 8 can be considered as a parameter characterizing the method of destruction. For example, an improvement in the destructive technique which leaves the fraction of prey destroyed unchanged but which causes a larger percentage of the predators to be eliminated would increase the parameter (3 and would not affect a; this would be reflected by an in- crease in 8. The parameter).. is a measure of the intensity of the destruction of both species but does not affect 8. The system of equations can now be written as
(13)
(14) The coefficient of the linear term in the last equation is always negative.
However, there are three possible alternatives for the linear coefficient in Eq. (13): (1) 0"1 - a).. >0, (2) 0"1 - a).. =0, or (3) 0"1 - a).. < O.
CASE 1: IfO"t =0"1 - a).. >0, we have the situation previously considered in section 3.2.1. The population sizes oscillate with period T.
CASE 2: If0"1 - a).. = 0, the system equations become
(15)
(16)
where (72" =(72+(710. After some manipulation one can obtain from Eqs.
(15) and (16r
(17) where C" is a positive integration constant. From Eq. (15) we see that N1 always decreases if neither of the population sizes is zero. And we see from Eq. (16) that N 2 has a maximum at N1 = (72"112, given Np ~ (72"112;
if not, N2 decreases to zero. We shall assume the interesting case where Np ~ (72"112. At N2 = 0, we have dNI/dt = 0, and the value of N1 corresponding to this extremal can be found from Eq. (17). It is simple to demonstrate that the two roots of this equation correspond to a maximum and a minimum ofN1 •
CASE 3:If(71 - Cl:,x <0, then it is apparent from the system equations that N1 constantly decreases to zero. N1 eventually reaches a value so small that it makes the positive term in Eq. (14) negligible; N2 then assumes a negative derivative and. so also goes to zero.
Ifwe put (71 - Cl:,x ~ -h with h >0, we can write
where C'II is a positive integration constant and (7;' =(72+{3'x.
The three preceding cases of behavior of prey N1 and predator N2 pop- ulation under the influence of external forces are graphically depicted in Fig. 1. The arrows along the curves indicate increasing time.
Even though the external forces applied to control the predator and prey populations affect them both, it is the magnitude of the effective co- efficient of autoincrease of the prey ((71 - Cl:,x) alone that determines the ultimate behavior of the association. In Case 1, the prey have recuperative powers which allow them to increase once the external forces have reduced the predator population size; as shown in Fig. 1, this leads to a cyclic as- sociation. If, however, the intensity of destruction is too great, the prey cannot increase even when the number of predators becomes very small;
this leads to the extinction of both species as shown in Fig. 1 by Curve III.
Case 2 depends upon the improbable situation that a combination of pa- rameters is exactly zero; perhaps this case should be judged as biologically implausible.
4. The Multispecies Lotka-Volterra Model
A generalization of the isolated and two species evolution equations leads to the so-called Lotka-Volterra equations for an arbitrary number of species m.
I
Figure 1. The three cases of the behavior of a prey N1 and a predator N2 under the influence of external forces: line I, (11 - Q'A > 0; line II, (11 - Q'A= 0 [al and bl are the two roots of Eq. (17)]; line III (11 - Q'A< 0 (From [1]).
dNi m
-d =Ni(bi+LaijNj), i=I,2, ...,m (18)
t . 1
]=
No mathematical constraints are posed on the various coefficients. The noted equations may represent either predator-prey or competition cases.
These nonlinear equations may exhibit rich dynamical behavior (even chaos with three equations or more). It turns out that sometimes the numerical solution of these equations may encounter difficulties, see discussion in [3J.
Therefore, a new semi-analytical solution is presented subsequently.