There exist two main problems in the analysis of correlation between population dynamics models with a discrete and continuous time space, which we consider ...
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Ecological Modelling 82 (1995) 93-97
Correlation between models of population dynamics in continuous and discrete time L.V. Nedorezov *, B.N. Nedorezova Institute of Medical and Biological Cybernetics, St.Ac. Timakov, 2, 630117 Novosibirsk, Russian Federation
Abstract
There exist two main problems in the analysis of correlation between population dynamics models with a discrete and continuous time space, which we consider at present paper. One problem is finding the discrete analog of the well-known Verhulst model. The other is the reason of appearance of chaotic regimes and oscillations with big periods in Moran-Ricker's models. Also we consider the new one-dimensional models with different assumptions about birth and death processes in populations. Keywords: Model comparison; Population dynamics
1. Introduction
In mathematical ecology there is the very wide-spread opinion that it is necessary to use differential equations for the description of population dynamics if its development is synchronized in time (Ricker, 1954; May, 1975; Varley et al., 1975; Williamson, 1975; Hassel, 1978; Vorontzov, 1978; Golubev et al., 1980; Bellows, 1981; Isaev et al., 1984; Smith, 1984; and others). Really, in a lot of different situations we can consider the qualitative and quantitative correlation between the theoretical curves which were obtained as solutions of systems of differential equations and experimental trajectories for real biological populations (Isaev and Khlebopros,
* Corresponding author
1973, 1977; Frisman and Shapiro, 1977; Sviregev and Logofet, 1978; Vorontzov, 1978; Isaev et al., 1984, 1988; Sharov, 1986). But in reality we have only a discrete development of the birth process and a continuous death process. It leads to the necessity of using models with a continuous time space, but trajectories of models have "jumps" at fixed time moments tk, k = 1, 2, 3 , . . . , tx+ 1 - t x - h = const > 0. t k is the m o m e n t of appearance of a new generation. Obviously all models which were constructed as systems of differential equations must be deduced from these models with discrete-continuous behaviour of trajectories (Nedorezov, 1986). When we consider a model with discrete time space we have no reasons to say that the realization of one or another dynamic regime is a result of special organized death or birth process. We have it because our knowledge about these processes is concentrated in a unique population
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L.V. Nedorezov, B.N. Nedorezova / Ecological Modelling 82 (1995) 93-97
p a r a m e t e r which is called "birth rate" and which equals the ratio of population sizes at nearest time moments. Respectively, the absence of this "interior interval population dynamics" leads to the appearance of different problems with the explanation of considering population dynamics, different dynamic effects etc. The typical way for constructing a model as any differential equation is following. We have a set of experimental points {x k} where x k is the population density at m o m e n t k = 0 , . . . , N. With the help of these points we calculate the values of birth rates at different moments Yk =Xk+l/Xk" After that we choose a monotonously decreasing function y ( x ) and determine its parameters under minimizing of the values of following function: N
Q = E (Yk - - y ( x ) ) 2 ~ m i n . k=l
After statistical analysis and using various criterions we obtain the following model of population dynamics:
Xk+ 1 = x k Y ( X k ) .
(1)
T h e r e exist a lot of different models which were constructed in this form (see, for example, Ricker, 1954; May, 1975; Golubev et al., 1980; Shapiro and Luppov, 1983; Isaev et al., 1984; Sharov, 1986; and others). After this sequence of mathematically based operations the illusion appears that we can use this kind of model not only for forecasting population dynamics but also for explanation of population dynamics effects. Moreover, it is easy to find in the literature many examples of deducing discrete models in the following way. The model with continuous time, for example, Verhulst's model dx at =X(al
-- a 2 -- ~ X )
(2)
where aK are the intensities of natural birth and death rates respectively and fl is a coefficient of influence of self-regulative mechanisms on the population size, one changes to the form:
x(t + at)
= x(t)(
l
-
x k= x(kAt),
where At is a time step. After the simplest calculations we obtain one of the well-known discrete models: 1 Zk+ 1 =
azk(1 --Zk),
a
(3)
Zk2 -- _ _ Xk2. Ol --0~ 1
Eq. 3 is called "discrete logistic model" and some authors assume that Eq. 3 is a direct discrete analog of Verhulst's model (Eq. 2). In both cases - when we deduce a discrete model from a model with a continuous time space, and when we obtain it under analysis of a set of experimental trajectories - we have a pure mathematical approach to population dynamic modelling and there is no real biological idea. The properties of model 3 are the properties of the differential approximation of an ordinary differential equation and we can not say that it has any relation to real biological population dynamics. If we want to construct real discrete models of population dynamics, we must use another way, which is based on equations with b r e a k functions in the right-hand side.
2. Models
Let's consider the non-parametric model of population dynamics which describes the changing of population size between two nearest moments of appearance of new generations: dx dt
-
xR(x),
x(0)=x0>0
(4)
where x ( t ) is a population size at m o m e n t t, R ( x ) is a death rate. R ( x ) satisfies to the following conditions R ( 0 ) > 0,
R ( ~ ) = ~,
dR -- > 0 dx
(5)
where R(0) is the natural death rate of individuals. In Eq. 5 we assume that population growth leads to an increase in death rate. Also we will assume that the model 4 - 5 describes the population dynamics only in the intervals A k = [t k, tk+l),
L.V. Nedorezov, B.N. Nedorezova/ EcologicalModelling 82 (1995) 93-97 and at fixed time moments t k we have the changing of the values of the population's size:
Xk=X(tk) = Yx(tk--O )
(6)
where x(t k - 0 ) is population size before the appearance of individuals of a new generation, Y is birth rate, which is equal to the number of new individuals per individual in the population. It is possible to point out the following three main variants of dependencies of Y from different arguments: 1. Y = const > 0; 2. Y = Y(x(t~ - 0)) or Y = Y(X~_l), which is used for the approximation of the real data; 3. Y = Y(£) where £ is the m e a n value of population size in the interval Ak: 1 2 = ~f~+lx(u)~j.
du.
tk
Theorem 1. I f Y = const, then in model 4 - 6 we have only the regimes of monotonous stabilization for values x k. The population does not become extinct if and only if the following inequalities is satisfying: g > e hR(°). Really, let 0(x) be the following function dx
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is a direct discrete analog for Verhulst's model (Eq. 2). Note that in both models we consider the similar dynamic regimes: monotonous stabilization at a unique stable level. In a partial case when R is a linear function and the birth coefficient Y = exp(--cx(tk -- O)), C -- const > 0, Eq. (4) reduces to the equation:
X~+l
a x k ( -- - exp 1 + yx k
a, b, y -= const > 0.
bxk ) 1 + yx k (8)
If y = 0, we have a M o r a n - R i c k e r model. Without losing the generality of the results we may assume that - / = 1. Theorem 2. Model B with y = 1 has the following properties: a. if a < 1, then x k ---)0 under k ~ ~ for all initial points x 0 > 0; b. if a > 1, then 0 is an unstable stationary state; ira > 1 and b < 1 or a > 1, b > 1 and a(b - 1 ) < be 1, we have the stabil&ation of the population at a unique level; c. let x* = a e -b, F ( x ) be the function on ,the right-hand side in Eq. 8; if a > 1, b > 1 and F ( x * ) > x*, then for x o > x* we have for all k, Xk>X*
= j xn(x) Obviously, 0 ( x ) is a monotonously increasing function: O'(x) > 0 and O"(x) < 0. It means that there exists the inverse function 0 - 1 , and we can reduce our model 4 - 6 to the difference equation
Xk+ a = Y t ) - l ( O ( x k ) -- h),
(7)
which has the monotonously increasing function in the right-hand side. Note that model 7 is a direct discrete analog for non-parametric model 4-6. For Verhulst's model (Eq. 2) (Verhulst, 1838) we have that Y = const and does not depend on the time m o m e n t t and population size x(t), R ( x ) = a 2 + Bx is a linear function. After simple calculations we obtain that Skellam's model (Begon and Mortimer, 1981; Sharov, 1986): ax k
xk+l-
l +bx k
a b-const>O,
The numerical analysis of the model 8 showed that if (a, b) ~ [0, 10]x[0, 10], we have the regimes of stabilization. When a and b are about 35, we obtained a stable 3-cycle. When a and b are about 92, we considered the fast decreasing of autocorrelation function and exponential growth of the distance between two different trajectories with near initial points. Obviously, if we have no self-regulative mechanisms in the population (i.e. when the death rate R - R(0)), we can obtain all discrete models of population dynamics of Eq. 1 type (for example, Hassel's model; Hassel, 1978) under the choosing Y in Eq. 7 of the second type. But for the larger part of real biological populations (and, especially, for the outbreak species of forest insects; Varley et al., 1975; Vorontzov, 1978; Isaev et al., 1984, 1988; Nedorezov, 1986) we cannot put this assumption because there is the strong influence
L.V. Nedorezov, B.N. Nedorezova / Ecological Modelling 82 (1995) 93-97
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of the self-regulative mechanisms on population dynamics (in particular, during the p e a k phase and phase of depression of the outbreak trajectory when we have a very high population density; Isaev and Khlebopros, 1973, 1977). Moreover, it is sufficiently difficult to obtain these well-known models with the discrete time space (Ricker, 1954; May, 1975; Frisman and Shapiro, 1977; Hassel, 1978; Begon and Mortimer, 1981; Bellows, 1981; Sharov, 1986) even in a most primitive case when R is a linear function. We can obtain these models in very rare cases when a specific relation between the birth and death rates exists. It means that we have some reasons to use these models for the description of population dynamics if and only if it has a sufficiently low density and the self-regulative mechanisms are very weak. On the other hand, practically we have no reasons to use these models for the description of outbreak species dynamics, of closed-living insects dynamics (Isaev et al., 1984, 1988) etc. where we have a strong influence of self-regulative mechanisms on population dynamics. Also it means that when we want to construct a model of Eq. 1 type and choose the dependence of birth rate Y from the population density, at the same time we choose the character of selfregulative mechanisms. But it needs additional investigation and, as a result, the formal way of constructing models of Eq. 1 type which we described in the Introduction, can not b e considered being correct. Of course, all models which were constructed this way can be used for forecasting population dynamics, but it is not possible to use these models for the explanation of population dynamics character or for the constructing of methods for optimal population exploitation. In a more realistic case when R is a linear function and Y = A e x p ( - y f f ) is of the third variant, A and y are positive parameters, ff is a m e a n value of the population size between two nearest birth time moments, we have the following model: Bx k
xk+l
( 1 + alx~) (az+a3xk)
B, 6, al, a2, a 3 - const > 0.
(9)
We can consider Eq. 9 as a generalization of Skellam's model (when p a r a m e t e r a t has a sufficiently small value), of Hassel's model (when parameter a 3 is small). When R is not a linear function, R = a + bx ~, where parameters a, b, v = const > 0 and Y is constant too, we have the following model: Bx k
Xk+l = (1 -t---a4Xk)V\l/u ,
B, a 4
-=
const > 0. (10).
Note that it is very difficult to construct the models 8-10 in a "direct way" without using the relations 4-7. These difficulties are in the necessity of explanation of the type of expressions on the right-hand sides of the models.
3. Conclusions It is necessary to point out two basic results. First, the realization of the chaotic dynamic regimes or the fluctuations with large periods in the well-known models with discrete time space (May, 1975; Frisman and Shapiro, 1977; Sharov, 1986; and many others) can be explained only as result of the special organized birth process in a population. If birth rate does not depend on the population values and time moments, we can have only the regimes of monotonous stabilization of population size at any level. And this behaviour of population size does not depend on character of self-regulative mechanisms. This general result showed us that the traditional way for constructing models of Eq. 1 type can be a good way for obtaining forecast models of population dynamics. But we have no reasons to use these models (with the exception of Skellam's model) for the explanation of dynamics effects, for constructing the methods of optimal population control etc. And second, Skellam's model (Sharov, 1986) is a direct discrete analog for Verhulst's model (Verhulst, 1838) of population dynamics. This is easy to find by direct simple calculations. In both models we have similar dynamic regimes of m o n o t o n o u s population stabilization at any
L.V. Nedorezov, B.N. Nedorezova / Ecological Modelling 82 (1995) 93-97
unique level. It is easy to find in the literature that Moran-Ricker's models (in particular, the "logistic model", Eq. 3) correspond to Verhulst's model. So, this last opinion we can not consider as correct. In this paper we considered only one-dimensional models of population dynamics. It is easy to show that it is possible to find discrete analogies for some other partial cases (for example, for models of dynamics of the populations with a sex structure, for the system of competition between two species etc.). But in many other cases (for example, in the case of constructing of the discrete analog of a predator-prey system) we must change the ideology in a creation of the model.
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