Simulation of certain oscillatory biological processes

The p r e m i s e t r e e is t r a n s f o r m e d in the following m a n n e r . L e t p be a s u c c e s s o r to P0. As a r e s u l t of a t r a n s f o r m a t i o n p" b e c o m e s a s u c c e s s o r to P0, P' b e c o m e s a s u c c e s s o r to p", and p b e c o m e s a s u c c e s s o r to p'. Other e x a m p l e s of t r a n s f o r m a t i o n s of the third group a r e o p e r a t o r s c o r r e s p o n d i n g to the application of the r u l e modus ponens to the f o r m u l a s of p r e m i s e s and c o r r e s p o n d i n g to the uncovering of definitions. Deductive t r a n s f o r m a t i o n s lead to the growth of both t r e e s . The deduction p r o c e s s is reckoned to be c o m p l e t e d if for each l e a f g in the t a r g e t t r e e t h e r e is a node in the p r e m i s e t r e e , whose f o r m u l a coincides with the f o r m u l a of g with an a c c u r a c y w h o s e l i m i t s a r e e s t a b l i s h e d by the substantive information of the nodes of the t a r g e t t r e e . F o r r e s o l v i n g the question on the a c c u r a c y of coincidence a s p e c i a l p r o c e d u r e has b e e n w o r k e d out, whose e s s e n t i a l f e a t u r e is the solving of p r o b l e m s in t h e o r i e s with one p r e d i c a t e , viz., an equality. If the p r o b l e m of e s t a b l i s h i n g deducibility in a f i r s t - o r d e r p r e d i c a t e calculus is solved with a~_ equMiky, then f r o m the t r a n s f o r m a t i o n s d e s c r i b e d above we can c o n s t r u c t a c o m p l e t e s e m i r e s o l v i n g p r o c e d u r e . We r e m a r k that when developing the m e a n s d e s c r i b e d above the possibility of using the a r c h i v e s of the m a t h e m a t i c a l text p r o c e s s i n g s y s t e m was allowed, for e x a m p l e , for seeking a u x i l i a r y s t a t e m e n t s , for unc o v e r i n g r e f e r e n c e s in the text. The m e d i u m and the deductive t r a n s f o r m a t i o n s operating in it a r e sufficiently u n i v e r s a l for solving p r o b l e m s f r o m v a r i o u s a r e a s using the tools of deductive inference. In the c a s e of solving p r o b l e m s f r o m individual sections of m a t h e m a t i c s the deductive m e a n s m a y be c o m p l e m e n t e d by c o n c r e t e transfornmtio~.s reflecting s p e c i a l methods of proof. LITERATURE 1, 2. 3. 4, 5o CITED V. M. Glushkov, "Some p r o b l e m s in a u t o m a t a t h e o r y and a r t i f i e a l intelligence," Kibernetika, No. 2 (1970). V. M. Glushkov and Yu. V. Kapitonova, "Automation of the s e a r c h f o r proofs of t h e o r e m s of m a t h e m a t i c a l t h e o r i e s and intelligent m a c h i n e s , " K i b e r n e t i k a , No. 5 (1972). V. M. Glushkov, K. P. Vershinin, Yu. V. Kapitonova, A. A. Letichevskii, N. P. Malevanyi, and V. F. K o s t y r k o , "On a f o r m a l language for writing m a t h e m a t i c a l texts," in: Automation of the S e a r c h for P r o o f s of T h e o r e m s in M a t h e m a t i c s [in R u s s i a n ] , Inst. Kibernetiki Akad. Nauk UkrSSR, I ~ e v (1974). F. V. Anufriev, " P r o o f s e a r c h a l g o r i t h m s for t h e o r e m s in logical calculi," in: Automata T h e o r y [in' R u s s i a n ] , Inst. Kibernetiki Akad. Nauk UkrSSR, KAev (1969). F. V. Anufriev and Z. M. A s e l ' d e r o v , " T h e c l e a r n e s s a l g o r i t h m , " K i b e r n e t i k a , No. 5 (1972i. SIMULATION OF BIOLOGICAL PROCESSES Yu. P. CERTAIN OSCILLATORY Yatsenko UDC 519.95 The m a i n c a u s e s of o s c i l l a t i o n s in the investigated biological s y s t e m s proved to be the specific nature of r e l a t i o n s between s p e c i e s (e.g., p r e d a t o r and prey), p r o c e s s e s due to age distribution, delays between v a r i o u s c o m p e t i t i v e p r o c e s s e s , and p e r i o d i c a l e x t e r n a l effects. Let us c o n s i d e r the f i r s t t h r e e c a u s e s , which a r e i n t e r n a l with r e s p e c t to biological s y s t e m s , u~ing models of biological s o c i e t i e s . Obviously, such a s e p a r a t i o n of the c a u s e s of biological oscillations is orAy r e l a t i v e l y t r u e since in r e a l s y s t e m s oscillations a r e caused by a combination of f a c t o r s including unkn~vn ones. On the other hand, s o m e of the indicated c a u s e s c a n o v e r l a p as i l l u s t r a t e d by the fact that delayed m a t u r ing of individuals in isolated populations t a k e s into account the c o m p e t i t i o n between individuals of various ages~ i.e., c a n s e r v e as a method to d e s c r i b e the effect of the population's age s t r u c t u r e . Brief Review of Models of Oscillatory Processes in Populations The f i r s t model of an o s c i l l a t o r y biological p r o c e s s is the V o l t e r r a model of the c o e x i s t e n c e of two antagonistic s p e c i e s , the s o - c a l l e d " p r e d a t o r - p r e y " model [1]. The model is d e s c r i b e d by a s y s t e m of two T r a n s l a t e d f r o m K i b e r n e t i k a , No. 5, pp. 108-113, S e p t e m b e r - O c t o b e r , 1978. Original a r t i c l e submitted May 15, 1978. 0011-4235/78/1405-0757507.50 9 Plenum Publishing Corporation 757 nonlinear d i f f e r e n t i a l equations with r e s p e c t to the f r e q u e n c y of p r e y N 1 and p r e d a t o r s N2: dM --i = dt dt Ni (e~ - - ~,tNJ, (t) ----- - - N~(~ ~l, e~, Yt, -- ~/~NO, Y~> 0. S y s t e m (1) has a s i n g u l a r point of the c e n t e r type: 9 ~'$ '~i ; (2) i . e . , undamped p e r i o d i c oscillations of the f r e q u e n c y of s p e c i e s take place around the values (2). V o l t e r r a [1] has found the o s c i l l a t i o n period, i n v e s t i g a t e d s m a l l fluctuations, and proved t h e i r i s o c h r o n i s m . The m a i n d e f i c i e n c i e s of the V o l t e r r a model a r e the nonasymptotic stability of the s t a t i o n a r y s t a t e of (2) and, as a r e s u l t , the s t r o n g dependence of s y s t e m b e h a v i o r on the p e r t u r b a t i o n of initial conditions: No m a t t e r how s m a l l the e x t e r n a l p e r t u r b a t i o n s , t h e i r effect is to t r a n s f e r the s y s t e m f o r e v e r into the other cycle. K o l m o g o r o v noted that the p r e s e n c e of periodic oscillations in the V o l t e r r a model is due to the choice of the f o r m of equation (1), and p r o p o s e d a different, much m o r e g e n e r a l , model for the d y n a m i c s of i n t e r acting p r e d a t o r and p r e y populations [2]: ~kt~= Kt (N,) Nt --L (N,) N 2, (3) dt w h e r e the functions on the r i g h t side s a t i s f y only g e n e r a l qualitative a s s u m p t i o n s . It has been proved that u n d e r c e r t a i n conditions p e r i o d i c oscillations (both stable and unstable) of the f r e q u e n c y of s p e c i e s can e x i s t and a limiting c y c l e is p o s s i b l e . To obtain a s y m p t o t i c stability of the s t a t i o n a r y s t a t e in the V o l t e r r a model (1), one has to c o n s i d e r other f a c t o r s that affect the d y n a m i c s of the f r e q u e n c y of s p e c i e s . One such f a c t o r is i n t r a s p e c i e s c o m p e t i t i o n (cons i d e r e d by V o l t e r r a in [1]} which t a k e s p l a c e in r e a l i t y when the supply of food is r e s t r i c t e d . In this c a s e the b r e e d i n g f a c t o r s of p r e y e 1 and p r e d a t o r s - 82 in equations (1) should be d e c r e a s i n g functions of population frequencies: el - XlN 1 and - - E 2 - - XzN2: The c o r r e s p o n d i n g s y s t e m of equations has a stable focal point: Nt - - etL~-~ e2~/t N 2 = Y2ei--Xtaz (4) S v i r e z h e v [3] p r o v e d that when i n t r a s p e c i e s c o m p e t i t i o n is sufficientlY s t r o n g the s i n g u l a r point (4) turns into a stable node, i.e., the oscillations of the s p e c i e s frequency vanish. An i n t e r e s t i n g g e n e r a l i z a t i o n of the V o l t e r r a model (1) has b e e n given in [3], w h e r e S v i r e z h e v suggested the following dependence of the decline function? on the f r e q u e n c y of prey: V (t) ~ I~N (t), !. [yiNi. 0 ~ Nt (t) ~ N~, , N t (t) ~ N 1, (5) which t a k e s into account the l i m i t e d c a p a b i l i t y of p r e d a t o r s to a b s o r b food. If N~ > e 2/Y2 the dynamic s y s t e m has a s t a b l e q u a s i l i m i t i n g cycle which includes the s t a t i o n a r y s t a t e (2) and is tangential to the line N 1 = N~. The c y c l e is s t a b l e with r e s p e c t to i n t e r n a l t r a j e c t o r i e s . Within the cycle the t r a j e c t o r i e s behave as in the $ model (1). If N 1 < e2/y2 all t r a j e c t o r i e s a s y m p t o t i c a l l y a p p r o a c h the line N2 = 0, which c o r r e s p o n d s to the c a s e of dying out of p r e d a t o r s as a r e s u l t of insufficient hunting activity. It should be noted that all m o d e l s of population d y n a m i c s using the tools of o r d i n a r y differential equations a r e only valid for t i m e i n t e r v a l s sufficiently long in c o m p a r i s o n with the longevity of individuals. They do not p r o v i d e an explanation f o r m a n y e x p e r i m e n t a l l y o b s e r v e d phenomena whose d u r a t i o n is of the s a m e o r d e r of magnitude as the longevity of an individual such a s , in p a r t i c u l a r , the oscillations of population frequency with such p e r i o d s [2]. It is thus n e c e s s a r y to take into account the age distribution. This r e q u i r e s the use of integ r a l and p a r t i a l d i f f e r e n t i a l equations. t T h i s function is c o n s i d e r e d to be l i n e a r in the V o l t e r r a model. 758 A model describing the dynamics single species has been given in [4]: of the age composition Ox of an isolated population of individuals of a ax 0-7-+ ~ = - - d (-c, t) x, (6) x (0, t) = ~ b (~, t) x (~, t) a.~ 0 with the initial condition x(r, 0) = ~(T). Here z is the age of individuals, x(r, t) is the age density, and b(r, t) a r e the b i r t h - and d e a t h - r a t e coefficients. It has been shown in [4] that the solution of (6) contains o s c i l l a t o r y components whose m a x i m u m period does not exceed twice the duration of the r e p r o d u c t i o n age. For the c a s e of a single breeding, the m a x i m u m period is equal to the r e p r o d u c t i o n age and the oscillations do not decay. The age s t r u c t u r e can also be taken into account on the b a s i s of o r d i n a r y differential equations. The population is divided into a finite number of age groups and equations a r e c o n s t r u c t e d d e s c r i b i n g the dynamics of each group and taking into account the t r a n s i t i o n of individuals f r o m group to group; this is equivalent to a d i s c r e t i z a t i o n of the model (6) with r e s p e c t to age 7. Such a d i s c r e t e model also allows oscillations of the f r e quency and age composition of the population [4]. In [5] continuous and d i s c r e t e models taking into account age distribution a r e used to explain oscillations in cell populations. L e t us now c o n s i d e r oscillations a s s o c i a t e d with delay. The f i r s t and m o s t general model of delay in biological societies is the V o l t e r r a " p r e d a t o r - p r e y " model with a f t e r e f f e c t [1]: dt "~ ei - - ~hN~- - Ft (t - - ~r N~ ('~)d'r Nt, -" (7) dN_..._~z= __%+ .~21f.i _~_ F2(t --'c)Nt(~:) d~ g~. dt Aftereffect indicates that the continuous sequence of past states of the s y s t e m (7) affects its future evolution. It has been shown in [1] that t h e r e exists a s t a t i o n a r y state of the s y s t e m : K,= ~ ,2-1- ~ F~('Od'~ 0 , K~= e~ , (8) ,l.-}- ~ F,('Od"r 0 such that s t a r t i n g at a c e r t a i n instant N 1 and Nz a s s u m e an infinite number of times the values K t and K~, i.e., execute infinite nonperiodic fluctuations around the s t a t i o n a r y state. S v i r e z h e v c o n s i d e r e d a different a p p r o a c h to the d e s c r i p t i o n of delay which takes into account o ~ y one past state of the s y s t e m [6]. A s s u m i n g that the effect of interaction of a given species with another on its f r e quency is delayed for the time T, the V o l t e r r a equation for n species turns into the following s y s t e m : Introduction of a delay r e d u c e s the s y s t e m stability and c a u s e s the o c c u r r e n c e of oscillations in c a s e s when no o s c i l l a t o r y solutions exist in the s y s t e m without delay. The a i m of the above b r i e f review of o s c i l l a t o r y p r o c e s s e s in populations was to indicate the causes of such oscillations and their m a t h e m a t i c a l d e s c r i p t i o n so that only the s i m p l e s t and most important r e s u l t s have been considered. It can be noted that different m a t h e m a t i c a l models c o r r e s p o n d to different oscillation causes. This c o m p l i c a t e s the study of the combined action of s e v e r a l f a c t o r s , whose i m p o r t a n c e has been s t r e s s e d in [41. Here we p r o p o s e a model for a " p r e d a t o r - p r e y ~ society which takes into account the effect of the roller,r ing f a c t o r s : a f t e r e f f e c t s in p r e y g e n e r a t i o n (i.e.~ the effect of age s t r u c t u r e of prey), aftereffects in the g e n e r a t i o n o f p r e d a t o r s by" p r e y , and the effect of p r e d a t o r s and p r e y on each other. It is to be noted that the "predaz t o r - p r e y " model with a f t e r e f f e c t as proposed by V o l t e r r a takes into account the last two factors only. 759 Model with of a Society of Two Antagonistic Species Aftereffect C o n s i d e r a biological s o c i e t y of two s p e c i e s one of which (the prey) s e r v e s as food for the other. The d y n a m i c s of this s o c i e t y will be d e s c r i b e d in t e r m s of the p r e y r e p r o d u c i n g i t s e l f and the p r e d a t o r a c c o r d i n g to the t w o - p r o d u c t m a c r o e c o n o m i c a l model of Glushkov [7]. His a p p r o a c h has been extended to n - p r o d u c t models of biological s y s t e m s with nonlinearities [8]. The d y n a m i c s of a biological s o c i e t y d i f f e r s f r o m that of industrial production d e s c r i b e d in [7] in that one has to c o n s i d e r two mutually independent instants of time: the instant the p r e y is b o r n u and the instant it is e a t e n by a p r e d a t o r ~-. The p r e y p a r t i c i p a t e s in the r e p r o duction of p r e y within the i n t e r v a l [u, T], and in the production of p r e d a t o r s in the i n t e r v a l [~-, + ~). In the i n d u s t r y , the tools of production p a r t i c i p a t e only in one kind of activity f r o m the instant of t h e i r creation. A c c o r d i n g l y , the model p r o p o s e d in [7] m u s t be s o m e w h a t modified. We introduce the following functions: N 1(t), the frequency of prey; N 2(t), the f r e q u e n c y of p r e d a t o r s ; m(t), the b i r t h r a t e of p r e y ; c(t), the b i r t h r a t e of p r e d a t o r s ; a(T, t), the n u m b e r of p r e y b o r n per unit t i m e at the instant t f r o m one p r e y b o r n at the instant T; fl(T, t), the n u m b e r of p r e d a t o r s b o r n p e r unit t i m e at the instant t as a r e s u l t of feeding on one p r e y at the i n s t a n t T; X(v, t), the p r e y s u r v i v a l function, i.e., the r a t i o of the n u m b e r of p r e y b o r n at the instant T and living at the instant t to the n u m b e r of all p r e y b o r n at the ins t a n t T (without c o n s i d e r i n g the c o n s u m p t i o n of p r e y by p r e d a t o r s ) ; # 0", t), the p r e d a t o r s u r v i v a l function [defined s i m i l a r l y to X(~, t)]; and y(t), the n u m b e r of p r e y c o n s u m e d at the instant t p e r unit t i m e r e l a t e d to the total n u m b e r of p r e y at the i n s t a n t t. The following c o n s t r a i n t s a r e i m p o s e d on the above functions: All functions a r e nonnegative, y(t) -< 1, ~(r, t) -< 1, # ( r , t) -< 1, X(T, t) =tl(r, t) = 0 for sufficiently l a r g e t - - r. The p r o p o s e d model c o n s i s t s of the following i n t e g r a l equations: m (t) = ~ ~ (u, t) (u, t) -- 0 (u, ~) y (~) m (u) du. c (l) = i' !~(% t) [~(x, t) y (T) t"~~.(u, z) m (u) dud% o 6 N~(t) (10) u i 0 is l ~(u, t) _I-- X(u, T)y('~)d~ m(u)du, u (11) (12) t N 2 (t) = I ~ (T, t) c (~) tiT, (13) 0 with r e s p e c t to the unknown functions m(t), c(t), Nl(t), N2(t), and y(t). The functions ~(u, t), 7,(u, t), p(u, t), and fl(u, t) a r e a s s u m e d to be given on [0, t]. F o r c l o s u r e of the s y s t e m (10)-(13) one has to specify the dependence of the r e l a t i v e r a t e of consumption y(t) on the functions m(t), c(t), N1(t), and N2(t). The s i m p l e s t f o r m of such a dependence is given by the Volt e r r a hypothesis: g (t)-- kN~ (t). (14) The age s t r u c t u r e of p r e y and p r e d a t o r s can be analyzed with the aid of model (10)-(14). F o r e x a m p l e , the n u m b e r of p r e y of a g e s between d 1 and d 2 is given by N'ra,'n,] -~" S X(u, t) 1 -- ~.(u, ~)g(~)d-c m(u)cht. (15) L e t us a s s u m e c e r t a i n s i m p l i f i c a t i o n s : The m o r t a l i t y and b i r t h r a t e of individuals depend only upon t h e i r a g e s v = t - u, the production of p r e d a t o r s by p r e y depends on t - T; i.e., ~. (~, t ) = X ( t - - % ~(.c, t ) = ~ ( t - - % e(.c, t)=o~(t--~), ~(z, t ) = ~ ( t - - x ) . 760 (16) M o r e o v e r , we a s s u m e that all individuals die only a f t e r r e a c h i n g a c e r t a i n age b 1 (prey) or b2 (predators); i.e.~ I1, t - - T . < b . x ( t - z ) - - - - I0, t--'~>/b~, (17) ~(t_z)___t 1, t--'~<b2, 10, t--~>~b2. Then, f r o m (10)-(13) we get the following s y s t e m of equations: N i (t) = t-S~:m (u) N2 (t) = 1 -- .~ y (~) d* du, i c ('0 a~, t~b2 (lS) m(t)~- ~ ( t - - u ) m(u) 1-- !'y(~s) d. du, '--b, c (t) = i ~ (t - - "0 Y ('~) L! m (u) dud*. max[O ~--btl t--b~ w L e t y(t) - 0; i.e., c o n s i d e r the c a s e of an isolated population of individuals of one s p e c i e s (pr~}). The population is d e s c r i b e d by the s y s t e m of equations re(t)= ( ~(t--u)m(u)du, t--b1 (19) N, (t) = i m (u) du. t--b, THEOREM I. For a (t - u) -- 5 0 system (19) allows weakly maximum period is close to the life span of individuals b v damped oscillations of m(t) and N1(t) whose In fact, s y s t e m (19) gives for this c a s e m" (t) = ~dn (t) -- aom (t -- b,). (20) Equation (20) has no purely periodic solutions [9]. Substituting into (20) a solution of the f o r m re(t) = Ce (5+i~?)t and solving the r e s u l t i n g equations for 5 and ~ a s s u m i n g that a0bl >> 1, we have in the f i r s t a p p r o x i m a t i o n 2.~k {1 - - ~--~-Oh)' ~= bi ~ (2ak)2 5k= bl(%g,,b,)~, (21) k=l,2 ..... S y s t e m (19) indicates that the f r e q u e n c y Nl(t) has the s a m e oscillations. QED. THEOREM 2. For a ( t - u) = ~05(t - u - T), w h e r e 5(x) is a delta function ( c o r r e s p o n d i n g to a single b r e e d i n g of individuals of age T), system (19) allows oscillations of m(t) and N~(t) with a maximum period equal to the b r e e d i n g age T. In this c a s e , f r o m s y s t e m (19) we have m (t) = %m (t - - T), (22) which has p a r t i c u l a r solutions of the f o r m e ( l n % / T ) t c o s ( 2 u k / T ) t . QED. Note that f o r an i s o l a t e d population of individuals of a single s p e c i e s , r e s u l t s s i m i l a r to T h e o r e m s 1 and 2 have b e e n obtained in [4, 5] on the b a s i s of the continuous a g e - d i s t r i b u t i o n model [Eq. (6)]. w C o n s i d e r now the full s y s t e m of equations which d e s c r i b e s the d y n a m i c s of the population of p r e y and p r e d a t o r s (18). L e t (23), 761 THEOREM 3. With assumptions (23), s y s t e m (18) has a stationary state 1 = 2 (~0b, - - 1) ~ ,. 1 eo= N Nto ~ kb2~ob~tzobi ' kb,b~=oO, ' 2 (aob i - - (24) 1) 2o ~ ' kbi%b, In fact, substituting Nl(t) -= const into the f i r s t two equations of s y s t e m (18) we get m(t) = eonst and c(t) - eonst. Substituting m(t) = m 0 and c(t) - c o into Eqs. (18) we obtain, a f t e r simple t r a n s f o r m a t i o n s , exp r e s s i o n s (24) for the s t a t i o n a r y state. THEOREM 4. The s t a t i o n a r y state (24) of s y s t e m (18) is unstable. To prove this consider the behavior of s y s t e m (18) n e a r the equilibrium position (24). Thus a s s u m e that re(t) = ~ l 2 (a~b, - - I) Zr 6re(t), c(t) ~- ~ ..... -t- octD, (25) w h e r e 6re(t) and 6c(t) a r e small quantities of the f i r s t o r d e r of s m a l l n e s s . Substituting (25) into (18) and neglecting t e r m s of the second o r higher o r d e r of s m a l l n e s s we obtain l i n e a r i z e d equations of s m a l l d i s p l a c e m e n t s of s y s t e m (18) in the neighborhood of the equilibrium position (24): 8m" ( t ) - o:oSm".(t) _[..2(% b,b~l)8m" ( t ) - - ( ~ o - - ~ )Sin" (t--b,) _---- . ~ pob~ [tic' (t) - - ~c" (t - - hi)] + l % [8c (t) - - 8c (t - - hi) - - 8c (t - - 02) + ~c (t - - b, -- bz)], blab ~ I 8c (t) + 8c"(t) - - b-~-- 2 (r176 - - 1 )Sin"(t--b,)~ (26) b2 8c (t - b2) b2 280 (~0b, - - 1) [tim(t) - - 8m (t - - bi) - - 8m (t -- b3) + 8tn (t - - bi - - b2)]. =ob~ An analysis of the c h a r a c t e r i s t i c quasipolynomial of the s y s t e m of differential equations with delay (26) gives a z e r o r o o t of multiplicity four, indicating that the s t a t i o n a r y state (24) is unstable. QED. It should be noted that the V o l t e r r a model [Eqs. (1)] which d e s c r i b e s the s a m e biological society without c o n s i d e r i n g delay is on the t h r e s h o l d of stability and has o s c i l l a t o r y solutions. The instability of the s t a t i o n a r y state (24) of s y s t e m (18) c o n f i r m s the hypothesis of S v i r e z h e v [6] that taking into account delay reduces the stability of population models. One can a s s u m e the existence of o s c i l l a t o r y solutions in s y s t e m (18). This probl e m is a n s w e r e d to s o m e extent by the following t h e o r e m . THEOREM 5. If b 1 = b 2 = b, s y s t e m (18) allows small oscillations of population frequencies around the s t a t i o n a r y state (24). To p r o v e the t h e o r e m we s e e k a solution of the l i n e a r i z e d s y s t e m of equations (26) in the f o r m e (6+i~)t, ~1 = 2~rk/b + e, w h e r e e and 6 a r e s m a l l quantities of the f i r s t o r d e r . Assuming that sob >> 1 and solving the r e s u l t i n g equations for 5 and e we have in the f i r s t approximation 2nk( 2 ) 6~=~ (2~k)~ th--~--~-- 1 -~ ~ka__4 , -2b(~/~__4) 9 (27) The m a x i m u m period of oscillations is a p p r o x i m a t e l y 0.75b. QED. Considering i n t r a s p e c i e s competition it is possible by p r o p e r s e l e c t i o n of the functions (16) also to obtain damped oscillations. It should be noted that within the scope of the proposed model it is possible to f o r m u l a t e optimization problems such as, for example, the m a x i m i z a t i o n of the f r e q u e n c y of p r e d a t o r s within the time interval [T1, T2] by choosing the function y(t). Similar p r o b l e m s of the m a c r o e c o n o m i c a l model of Glushkov [7] have been discussed in [10]. LITERATURE 1. 762 CITED V. V o l t e r r a , Mathematical Model of the Struggle for E x i s t e n c e [Russian translation], Nauka, Moscow (1976). 2. 3. 4. 5. 6. 7. 8. 9. 10. A.N. Kolmogorov, "Qualitative studies of mathematical, models of population dynamics," in: Problems of Cybernetics [in Russian], No. 25, Nauka, Moscow (1972). Yu. M. Svirezhev, "Mathematical models of biological societies and control and optimization problems associated with them," in: Mathematical Simulation in Biology [in Russian], Nauka, Moscow (1975). R.A. Polu~ktov (editor), A Dynamic Theory of Biological Populations [in Russian], Nauka, Moscow (1974). Yu. M. Romanovskii, N. V. Stepanova, and D. S. Chernavskii, Mathematical Simulation in Biophysics [in Russian], Nauka, Moscow (1975). Yu. M. Svirezhev, "Vito Volterra and modern mathematical ecology," in: V. Volterra, Mathematical Model of the Struggle for Existence [in Russian], Nauka, Moscow (1976). V . M . Glushkov, "On class of dynamic macroeeonomical models," Upr. Sist. Mashiny, No. 2 (1977). V . M . Glushkov, V. V. Ivanov, and V. M. Yanenko, Simulation of I n t r a - a n d Intercellular Interactions Based on a Class of Dynamic Macromodels [in Russian], Preprint, Inst. Kibernetiki Akad. Nauk UkrSSR$ No. 78-71, Kiev (1978). L . E . El' s g ol 't s , Introduction to the Theory of Differential Equations with Deviating Arguments, HoldenDay (1966). V . M . Glushkov and V. V. Ivanov, "Simulation of the optimization of workplace distribution between production branches A and B ; Kibernetika, No. 6 (1977). ANALITIK-74 V. M. G l u s h k o v , T . A. G r i n c h e n k o , A. A. D o r o d n i t s y n a , A. M~ D r a k h , Y u . V. K a p i t o n o v a , V. P. K l i m e n k o , L. N. K r e s , A. A. L e t i e h e v s k i i , S. B. P o g r e b i n s k i i , O. N. S a v c h a k , A. A. S t o g n i i , Y u . S. F i s h m a n , a n d N. P . T s a r y u k UDC 681.3o06.51 1. I N T R O D U C T I O N The algorithmic language ANALITIK-74 is an input lan~dage for type MIR-3 computers for the automation of engineering and scientific calculations, hence its orientation on the description of algorithms that implement numerical and analytic methods of solution of engineering problems~ The overwhelming majority of intended users of both the machine and language a r e not professional mathematicians or p r o g r a m m e r s but engineers and scientists with rat her limited training in programming. Thus, efficient use of the machine requires a language resembling as closely as possible the language of mathematical disciplines associated with mathematical analysis (differential and integral equations, mathematical physics, linear algebra, mathematical statistics, etc.). This explains the abundant mathematical sy m bolism used in the language. Of p r i m a r y importance for engineering problems is the capacity of the language to describe calculations in t e r m s of various numerical algebras such as the following: a) the algebra of whole and rational numbers necessitated, first of all, by the fact that many t ran sfo rmations ar e applied only to the field of rational numbers and that in this algebra it is possible to obtain results with an absolute accuracy and frequently in a most compact form; b) algebras of real numbers of any dimensionality that reduce the accumulation of round-off e r r o r s thus allowing one to extend the applicability of numerical methods; c) algebras of complex numbers in which besides the basic operations (addition, multiplication, subtraction, and division)all standard functions (sin, arcsin, sinh, in, exp, etc.), also a r e defined~ Translated f r om Kibernetika, No. 5, pp. 114-147, September-October, 1978. Original article submitted March 27, 1978. 0011-4235/78/1405- 0763507.50 91979 Plenum l~blishi~g Corporation 763