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LIMIT CYCLES OF LENGTH TWO IN THE RIKKER MODEL AND THEIR APPLICATION IN FISHING
1 GuRAMi Tsitsiashvili, 2 Tatyana Shatilina, 1,3 Marina Osipova,
1 Tatyana Radchenkova
1 Russia, 690041, Vladivostok, Radio street 7, IAM FEB RAS, 2VNIRO (TINRO), 690091, Vladiv ostok, lane. Shevchenko, 4 3 Russia, 690002, Vladiv ostok, FEFU Campus 10 Ajax Bay, Russian Island [email protected] o.ru, taty ana.shatilina@tinr o-center.ru, [email protected], [email protected]
Abstract
The paper investigates the limit cycle of length two in the Rikker model. It is established that the dependence of the ratio of the maximum value of the cycle to the minimum depends monotonously and almost linearly on the growth coefficient of the Rikker model. Models of the parity shift of the limit cycle of length two are constructed, which is provided by a simultaneous sharp decrease/increase in the growth coefficient. On the example of the Amur salmon in 1994 It is shown that a decrease in the growth coefficient, leading to a shift in the parity of the cycle of length two, is accompanied by a low temperature during the life cycle of pink salmon, when the pink salmon population is in a state of spawn and when the young are rolling.
Keywords: the limit cycle of length two, the parity shift of the cycle and its conditions, the growth coefficient of the Rikker model.
1. Introduction
In chaos theor y, and specifically in population dynamics, the Riker model is a population gr owth model [1]. In ichthyology research, it aroused the interest of mathematicians in determining the length of limit cycles depending on the value of the growth coefficient (Malthusian parameter) [2] - [5]. Despite numerous studies of the Rikker model, the analysis of limit cycles of length two may lead to new results.
In this paper, the ratio of the maximum value of the limit cycle of length two to the minimum is investigated. With the help of computational experiments, it is shown that this ratio depends monotonously and almost linearly on the growth coefficient for almost the entire range of values. This result may be applied to solving an important applied problem formulated by A.A. Goryainov, an employee of the Tinro Center [6], [7] on changing the parity of the limit cycle of length two in the Rikker model. If the maximum value of a cycle of length two is taken in even (odd) years, then a change in parity is understood as such a change in the cycle in which the maximum value begins to be taken in odd (even) years.
The task of changing the parity is important for predicting significant changes in populations, the dynamics of which is subject to the Rikker model. Such parity changes occur very rarely and are the result of a significant influence of external (hydro meteorological) conditions on population dynamics. A featur e of the method for solving the problem of parity change is the consideration of the Rikker model at an extreme value of the growth coefficient, which makes it possible to determine the conditions for parity shift from hydro meteorological data. It is
shown that the very shift of the cycle of length two at the minimum point of the cycle leads to a significant decrease in catches. On the contrary, at the maximum point of the cycle, a very high growth coefficient is required for its implementation. These circumstances allow us to point out an analogy betw een the parity shift (number/catch of pink salmon) and the failur e of the technical system.
2. Methods
Consider the Rikker model xn+1 = flnxn exp ( — bxn ), n > 0. Using the standard substitution yn = bxn, we arrive at a recurrent sequence
yn+1 = f (yn) = anyn exp (-yn), n > 0. (1)
Here an = fln/ b is the growth coefficient and cn = yn+1/ yn is the return coefficient. It follows from the formula (1) that the equality an = cn exp (yn) is fulfilled, linking the growth coefficient an with the return coefficient cn and with yn.
Let's focus on the case when an = a. In this case, the following classification of stable limit modes in the Rikker model [1] - [4] is known. For 0 < a < = 1, the sequence yn, n > 0, has a stable rest point Y1 = 0. At < a < « e2 « 7, 39 the sequence yn, n > 0, has a stable rest point Y2 = Y2 (a) > 0. At fi1 < a < « 12.49 the sequence yn, n > 0, has a stable limit cycle of length two. At j82 < a < « 14.68 the sequence yn, n > 0, has a stable limit cycle of length four, etc.
Let's calculate the components of the limit cycle of length two Y, f (Y), defined for Y > 0 by the relations
Y = f (f (Y)) ^ 1 = a2 exp (-Y(1 + ae-Y )) ^ <p(Y) = ^(Y), (2)
<p(Y) = 2 ln a, ^(Y) = Y(1 + ae-Y).
Numerical calculations of the roots of the equation (2) show that for < a < this equation has three roots, because the function ^(Y) has both a minimum and a maximum. Moreover, the minimum root Ym,n and the maximum root Ymax are related by the relations Ymsx = f (Ymin), Ymin = f (Ymax). The root Ym,d = ln a, contained between the minimum and maximum roots, corresponds to the unstable rest point of the sequence yn, n > 0, (see Fig. 1).
Figure 1. Graphs of functions f(Y) (red line), ^(Y) (blue line) at a = j82. The results of calculating the roots of the equation (2) are presented in Table 1.
Table 1. The values of the roots , Ymin, Ymax and the ratio Ymax/ Ymin depending on the
growth coefficient a, fii < a < fi2.
a Y / Y ■ 1max' 1min Y ■ 1 min Ymid Y 1 max
7.39 1.02807 1.97244 2.00013 2.02781
8 2 1.38629 2.07944 2.77259
9 3 1.09861 2.19722 3.29584
10 3.92745 0.934596 2.30259 3.67057
11 4.83672 0.821659 2.3979 3.97413
12 5.74133 0.737215 2.48491 4.2326
12.15 5.87697 0.726287 2.49733 4.26837
12.3 6.01263 0.715737 2.5096 4.30346
12.45 6.14831 0.705543 2.52172 4.3379
12.49 6.1845 0.702882 2.52493 4.34697
From the table 1 it can be seen that the ratio Ymax/ Ymin increases with the growth coefficient a increases from a value close to one at a ~ to a significantly larger unit (« 6.1845) value at a ~ j82. This is shown in more detail in Fig. 2.
Figure 2. Graph of the dependence of Ymax/ Ymin on the growth coefficient of a at fii < a < fi2.
As a result, it has been empirically established that on the segment fii < a < the ratio Ymax/ Ymin depends on the growth coefficient a almost linearly (except for a small tail on the left). Moreover, this fact cannot be established analytically .
Using these estimates and formulas (1), (2), we give one example of the parity shift of a stable cycle at the minimum point (see Fig. 3). Let an = j82, 0 < n < 9, n = 4, a4 = fi*2 = exp (0.702882 ) « 2.01956, and the sequence yn, 0 < n < 9, n = 4 coincides with a stable cycle of length two y0 = y2 = y4 = y5 = y7 = y9 = 0, 702882, yl = y3 = y6 = y8 = 4, 34697. However, at n = 4, ther e is a shift in the parity of the stable cycle.
Figure 3. The graph of the parity shift in the Rikker model at the minimum point.
This calculation gives some idealized example of the parity shift of a cycle of length two. To achieve such a shift, it is necessar y at the moment n = 4 to signific antly reduce the growth coefficient, namely, by j82/ ft* « .6, 184 times.
In turn, the graph of the parity shift of the limit cycle of length two at the maximum point is possible at the point n = 3 at a3 = fi*2* = exp (4, 34697) « 77, 244. Let an = 0 < n < 9, n = 3, a3 = p2*, and the sequence yn, 0 < n < 9, n = 3 coincides with a stable cycle of length two: y0 = y2 = y5 = y5 = y7 = y9 = 0.702882, y! = y3 = y4 = y6 = y8 = 4.34697. However, at n = 3, a steady cycle parity shift occurs. To achieve such a shift, we need to greatly increase the growth coefficient, namely, by ^2*/ « .6,184 times.
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Figure 4. The graph of the parity shift in the Rikker model at the maximum point.
Thus, in the idealized parity shift model of the limit cycle of length two, it is necessar y to significantly (6,184 times) simultaneously reduce/incr ease the growth coefficient. Therefore, a more realistic procedur e is to reduce the growth coefficient and the corresponding shift of the limit cycle of length two at the minimum point.
3. R esults
Let's focus on the meteor ological conditions that ensur e a parity shift at the minimum point of the limit cycle for the dynamics of the Amur pink salmon population. Similar studies have previously been conducted for the Seaside pink salmon [8]. Detailed ichthyology studies show that negative meteor ological effects leading to a parity shift are possible in two periods of the pink salmon life cycle. The first period is incubation: in Januar y, when the pink salmon population is in a state of caviar. The second period is in June-July during the decline of young pink salmon. Table 2 shows data on the air temperatur e in Nikolae vsk on Amur in the first period and in the second period on the water temperatur e in the Tatar Strait. These data show that the parity shift of the length two cycle occurs at sufficiently low temperatur es during the specified periods of the Amur pink salmon life cycle.
Table 2. Temperatur e data for 1994.
Month HMS Air Temperatur e Water Temperatur e in
Nikolae vsk-on-Amur Tatar Strait
Januar y -6.4 0.5
Februar y 3.3 0.3
March 0.1 0.3
April -0.8 0.1
May -2.2 -0.6
June -1.5 -0.9
July -0.9 0.1
August 1.2 0.9
September 0.5 0.2
October 0.9 0.2
November -0.8 0.1
December -2.8 0.5
4. Discussion
The problem of shifting the parity of a cycle of length two on the one hand is of serious theoretical and practical interest. When solving it, we have to limit ourselves to cycles of length two in order to compress the initial biological and hydrometeor ological information. Of course, this appr oach to analyzing the sour ce infor mation allo ws for a certain appr oximation. But such an appr oximation can be justified by setting the problem of shifting the parity of a cycle of length two. Moreover, when solving this problem, it is necessar y to analyze in detail the life cycle of the Amur pink salmon and identify critical moments in it.
5. Conclusion
In conclusion, it should be noted that the idealized model of the parity shift of a cycle of length two is not alw ays implemented in practice. The parity shift can occur in states wher e the ratio of the maximum cycle value to the minimum value is close to one. Nevertheless, even in this case, there is a decrease in the growth coefficient caused by adverse meteorological conditions.
R eferences
[1] Rikker,U. E. (1979) Methods of assessment and interpretation of biological indicators offish populations. Mosco w: Food Industr y, 408 p. (In Russian).
[2] Shapir o, A. P., Luppo v, S. P. (1983) Recurr ent equations in the theory of population biology. Moscow: Nauka, 132 p. (In Russian).
[3] Lasunsky , A. V. (2012) On the period of solutions of a discrete periodic logistic equation Proceedings of the KarSC RAS, vol. 5. Ser. Mathematical Modeling and Information Technology , No. 3, pp. 44-48. (In Russian).
[4] Elaydi, S. N., Luis, R., Oliveira, H. (2011) Towards a theory of periodic difference equations and its application to population dynamics. Dynamics, Games and Science. Springer Proc. Math. Vol. l, pp. 287?32l.
[5] Shlufman, K. V., Neverova, G. P. , Frisman, E. Ya. (2017) Dynamic modes of the Riker model with a periodically changing Malthusian parameter . Nonlinear dynamics. Vol. 13. No. 3, pp. 363-380. (In Russian).
[6] Goryainov, A. A., Shatilina, T. A. (2003) Dynamics of the Asian pink salmon and climatic changes over the Asia-Pacific region in the twentieth century. Marine Biology. Vol. 29, No. 6, pp. 429-435. (In Russian).
[7] Goryainov, A. A., Krupyanko, N. I., Shatilina, T. A. (2013) Comparativ e analysis of catch dynamics of the Primorye and Amur pink salmon Bulletin No. 8 of the study of Pacific salmon in the Far East. Vladivostok: TINRO Center, pp. 106-118. (In Russian).
[8] Lysenko, A. V., Shatilina, T. A., Gaiko, L. A. (2021) The influence of hydrometeor ologi-cal conditions on the dynamics of catch (abundance) of the Primorsky pink salmon ON-CORHYNCHUS GORBUSCHA (SALMONIDAE) based on retrospectiv e data (Sea of Japan, Tatar Strait). Questions of Ichthyology, Vol. 61. No. 2, pp. 206-218. (In Russian).
