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V. Mikhailova v. 3 of 22000 doctor of technical sciences, professor, [email protected]

A. Y. Perevaryuhaa, ph.D. Senior researcher, [email protected]

Yu. S. Reshetnikovb, doctor of biological sciences, professor, [email protected]

A st. Petersburg institute of informatics and automation, russian academy of sciences, 199178, v.O. 39.14 liniya, st. Petersburg , russian federation

Ba. Institute of ecology and development. N. Severtsov russian academy of sciences, 117071, moscow, lenina pr. 33

Introduction. Populations of commercial fish species develop in aquatic ecosystems according to the internal mechanisms of their evolutionary adaptations, which do not always correspond to environmental conditions, which can change under the influence of anthropogenic impact. Often, the factors of eutrophication of water bodies or the introduction of previously unknown species into the ecosystem exacerbate the competitive confrontation of populations. Changes propagate like dominoes and complicate the development of a strategy for the rational use of biological resources. Species under suboptimal anthropogenic conditions are vulnerable. The task is to release a model of the introduced population dynamics, implemented as a group of scenarios for a changing environment. Results: a computational model of the population has been developed to describe scenarios for its adaptation to environmental conditions. Scenarios include the rate of size development as fish food. The model includes a block for calculating diets taking into account the factors of the hydrological situation: oxygen content and energy of hydrogen ions. The model is able to teach in two modes: under standard environmental conditions (without specifying the hydrological situation) and under given conditions of anthropogenic changes. Our approach made it possible to predict changes in the architecture of the population under the variability of abiotic factors. The model demonstrates the risk of catching the fish that contribute the most to the rate of biomass growth. Practical significance: the model was identified using information about the abundance of whitefish in lake sevan. It is suitable for computer simulation experiments, which made it possible to describe the features of population dynamics scenarios for cases of increased fishing, limited feeding, and changes in the hydrological conditions of the lake.

Keywords - nonlinear population models, fish growth models, calculation energy balance, commercial exploitation scenarios, eutrophication.

Articles

Citation: mikhailov vv, perevaryukha a. Yu. Reshetnikov yu. S. Model of the dynamics of the fish population with an assessment of the individual growth rate and scenarios of the hydrological situation. Information and control systems. 4, pp. 31-38. Doi:10.31799/1684-8853-2018-4-31-38

In our previous work [1], we used mathematical methods to analyze the dynamic balance of the main nutrients that form the first level of the ecosystem. A computational model of the consequences of excessive formation of nitrogen and phosphorus in the bottom sediments of a large reservoir has been developed. As more feed compounds enter the reservoir as a result of human activities, eutrophication processes begin (algal blooms and lack of oxygen). We successfully parametrized the model according to the actual data of the reservoir and calculated the forecast of hydrochemical parameters. Ecosystems are very susceptible to the domino effect. The transformation of the trophic status of a reservoir from oligotrophic to eutrophic after a certain period of time is also expressed in its aquatic community. Species adapted to urban conditions cannot change rapidly, therefore, preference is given to active unwanted invaders that are less demanding on low oxygen content.

Equations with a redefined computational structure for describing the efficiency of sturgeon reproduction based on various survival factors at the stages of development of juveniles. For an exploited population with adaptive cyclicity, we described the calculated scenario of degradation in the form of a sudden collapse. This article is a logical continuation of our research in the field of mathematical biology. This is an aggregated model of the dynamics of the fish population, related to environmental conditions according to the accepted hydrological indicators through the efficiency of biomass accumulation and the rate of linear development, taking into account unbalanced nutrition. Estimates of the effectiveness of replenishment of stocks are often unpromising with a rapid change in the trophic state of the reservoir

Shift. We took into account possible methodological errors of mathematical analysis [three quarters].For reference, whitefish are typical representatives of the arctic ichthyofauna [5-9], so their presence in the middle latitudes is possible only in high-mountain lakes, such as issyk-kul, sorulukel or sevan.

Characteristics and components of the model

The structure of the population model combines three submodels: 1) model of individual weight dynamics; 2) a model of the dynamics of the number of age groups; 3) a model of the dynamics of the release of eggs with a constant calculation of the loss of fry. These submodels are adapted for joint calculations, but they can be used independently for solving one-time problems, for example, analyzing data on the energy of food and respiration of fish (you need a case of eutrophication of a reservoir) or units of coefficients of natural and commercial mortality by age groups of fish.

The interaction of submodels is organized in such a way that films can describe important intrapopulation mechanisms: dependence of diet and weight on the volume of fish ages; dependence of the yield of fish caviar on the diet; dependence of natural mortality rates on the actual diet; dependence of natural mortality rates on spawning game; dependence of fecundity on the abundance and age of fish. The parameters of the standard environmental conditions are set in the initial state settings.

We accept that abiotic factors directly affect two population indicators: fish diet and mortality of their eggs; and indirectly can affect the abundance of individuals, their mortality and fertility, which has been accumulated on the example of whitefishes [4, 10, 11]. In our model, the indistinct temperature and the concentration of dissolved o2 shape the amount of food and egg mortality; the predominant influence on egg mortality is the ph of the water.

The general algorithm of our virtual kitty lies in the fact that initially for all ages, like the original news about individual weight, we determine the maximum amount of food, breathing costs and growth mass of fish corresponding to the maximum amount of food. Then, based on the scenario data, the actual volume of forces received by the fish of the age group is calculated, and the weight value is adjusted taking into account the diet. The next step is to calculate the number of age groups after registration of natural mortality and commercial withdrawals. It is assumed that all fish that have reached a given weight by the hour of spawning participate in reproduction, and the number of eggs is proportional to the size and years of spawning fish.

Units of calculation for individual indicators

Maximum amount of food and body weight gain of fish is determined for a month, subject to the exceptional happiness of their nutritional needs. The internal time step of the unit is considered equal to days. Since the mortality of juveniles is extremely high, the calculation of the daily amount of food is done according to their actual number, taking into account daily mortality. For other ages, the amount of food is calculated based on a fixed number in any month. The gai-nis weight is calculated as follows:

Wm (t 1) = w (t) awm(t 1), awm (t 1) = cm x u - r,

Where w, wm are the real and maximum weight of the fish; awm is the maximum gain; cm - the maximum amount of food; u is the coefficient of assimilation of food; r - breathing costs. The standard amount of feed corresponds to the average monthly amount at which the fish gain a given standard weight. Based on biological data [12, 13], we determine the amount of food as a power-law function of weight:

Cm =a0 xf ai,

Where cct1 is the feed rate for fish weighing one kilo; а0 - correction for excess of the maximum amount of feed over the standard; a1 e-1/2, two-3]; cm is one of the sides of a food item for fish weighing w kg.

The normative amount of food is calculated depending on the average monthly temperature of vagueness and the oxygen content in the liquid: cct1 = a2xatxak, where a2 is a constant coefficient, at , ak are corrections for temperature and oxygen levels.

The dependence of food volume on temperature is non-linear; if the weather is very favorable for a given fish species, the curve has a maximum. If the temperature reaches the limit, the amount of food decreases to negligible values, and eating stops. In the model, the dependence is approximated by the following formulas:

0 ^Opt T > tmax ,t Let suppose that in the situation when the oxygen concentration is greater than the given limit value o2 > o2 lim, this factor is no longer the limiting factor. During the reduction of oxygen in water during eutrophication, the amount of food first decreases rapidly from normal to zero,1 and then gradually decreases

Down to zero.The dependence is approximated in the model as follows:

C^min 2 s

X (o2 -02min ag (o2 -02min) - ag (o2 -c^min) ;

02 Calculation of energy expenditure is based on the assumption that breathing expenditure includes pairs of components that include maintenance costs ichthyomass and the cost of its growth according to the formula r =a(w) x (rn - rp).

The brett and groves scheme [14], which determines the ratio of active and standard metabolism depending on temperature. Information on the increase in the ratio of active and standard metabolism, as well as the distribution of the efforts coming with food on biomass and respiration in fish ontogenesis were taken into account [15, 16].

It was established that that maintenance costs are proportional to biomass, increasing with increasing temperature increases in accordance with the krogh function: rn = k(t) x w xa16, where k(t) is the krogh function, a16 is the empirical coefficient. The cost associated with growth is proportional to the amount of food, and decreases linearly with increasing temperature according to the law rp = c(1 - a17) x a18.

Food spectrum calculation unit. Model access to identification uses facts about food biomass and priorities of its consumption by age groups of whitefishes. The data work here is the average annual changes in the food spectrum depending on the age of the fish. The device calculates the actual amount of food for fish under the available food biomass, as well as with requests for an important type of food from all competing whitefish ages [17]. The average monthly biomass of various types of food is presented in proportion to their share in the diet, which guarantees a constant tension in food relations. The absolute value of feed biomass is given by a periodic function to provide the standard amount of feed for fish at a normal level of their nutrition (average % of the average monthly biomass). Let us recall that a sharp change in conditions causes whitefish to drastically change their food spectrum, up to usually turning into predators [18–20].

Algorithm for correcting biomass growth. An iterative algorithm adjusts biomass gain and respiration costs to the actual diet:

W(t 1) = w(t) awm (t)x c/cm ;

R(t 1) = rm (t l)x s/cm.

The calculations assume that the food distribution structure does not depend on the amount of food. Separately, a weight correction is introduced after spawning for age groups that take part in reproduction.

Calculation of natural mortality rates.

A very important component of modeling is the model loss of biological resources. The natural mortality curve is u-shaped. Mortality in the first age group is high, then decreases, reaching a minimum before spawning, and increases again, approaching a maximum at the 10th year of life of the whitefish [5, 21]. Let us assume that the mortality of fish depends on the degree of satisfaction of their nutritional needs. Losses due to lack of food can be modeled as follows:

C/cm > kg, kg = 0; — mortality rate dependent on starvation, kg — starvation threshold. In the exponential factor, we take into account the relationship between mortality and relative monthly growth. The natural mortality rate kk in general for groups can be found using the following formula:

Kk = 1 - (1 - kkst) x (1 - kkner) x (l - kkg), 

Where kcct is mortality without spawning under normal feeding conditions; kcner - loss during spawning. We have to calculate the losses associated with density and competition.

Method for estimating the efficiency of reproduction

Let's determine separately the estimate of losses near. Stages of development, which would additionally depend on the initial density of eggs .N(0). The well-known ricker function r = n(0)exp(-bn(0)) is hardly suitable here, because the iteration trajectory exhibits the properties of chaotization with increasing e, > e2, which serves as a regime inexplicable for biology [2]. ]. For our calculations, we will use the implementation of a continuous segment in iterative model building. In this article, we apply a modification of our replenishment model to shorten the number from n(0) to r = n(t) like a system of equations over a time interval that can be continuous vulnerability interval t e [0, t]:

Dd^ = -(у w(t)n(t) ©(s)p)7v(i); d w _ g

, ©(S) = (l - )_1,

Y is the mortality rate depending on the population density; p is the neutral loss coefficient; w(t) is an indicator of individual development in the initial ontogeny under the influence of group crowding. ©(S) reflects the decrease in spawning efficiency for a small population after overfishing: lim ©(s) = 1.

Numerical solution of the cauchy problem (*) will calculate the loss from eggs to fry, according to the dependence from a pronounced maximum, which will be the most effective size of the spawning group.With an increase in the size of the reserve, the efficiency of reproduction decreases, albeit also with a non-zero horizontal asymptote.

Only the use of empirical dependencies for any character, in tabular form, makes it possible to provide predictive calculations of the actress with the goal of studying the survival of eggs, taking into account changes in parameters environment. The death of eggs, in addition to predation, is associated with the temperature of vagueness and the amount of oxygen dissolved in water. For the reservoir eutrophication scenario [5], corrections can be used that are based on the sequence of calculating the oxygen content in the nutrient balance model described in the modern previous article.

When calculating the total number of eggs n(0) for system of equations (*), it is considered that the fish takes part in spawning, if by the time of spawning it will be mass wner. The number of eggs n(0) is determined in proportion to the mass and age of the fish: n(0) = a20 a21 x w(l) a22 x l, where l is the age of the spawning fish; w is the weight of the fish, w > wner. In the event that the fish gains weight more than wner at the appointed time of spawning, it can spawn again; otherwise, it must skip spawning.

The balance of the number of ages is calculated after each month, taking into account natural mortality and the share f e [ 0, 1] of fishing (which may be legal and unrecorded): n(t 1) = = n(t) x (1 - (1 - kc) x (1 - f). Finally, the iterative algorithm calculates the transition of fish to the next age group and the replenishment of the population with fish that survived the juvenile stage, when the data are most vulnerable. Have natural enemies.

Identification population model

The population model was determined based on data on the whitefish population in lake sevan, armenia. This whitefish is a hybrid of core-

Gonus lavaretus maraenoides and coregonus lavaretus ludoga, introduced into lake sevan in the 1920s after their natural hybridization, many morphological and biochemical parameters in our model, as a single population [22], are long after introduction ladies did not play a big role in the fishery, however, since the 1960s, their population has increased, well, in the 1980s they became the most important fish for local fisheries. The intensification of fishing has changed the age structure of the population, and now there are few fish over 8 years old. Exactly with the one you chose, the maximum age of the fish in the virtual pussy being performed is listed as 10 years old. The average weight of fish in catches is 840 g at the age of three years. The available literature data quite fully describe the food spectrum of fish from lake baikal. Sevan [23]. Our specialists were able to determine the daily diet of whitefish sr by calculating the bar according to the system of a. V. Kogan [24]. It turned out to be equal to 4.3% of the set of the fish body. The daily rate of satiety is 1/4 of the daily diet. Taking into account nutrition, depending on the water temperature, a table of annual changes in the diet was compiled. Somewhere up to 12 months, the daily ration forms 1.9% of the fish weight, and the annual ration is about 7 fish weights. Similar values describe the nutrition of whitefishes in the scheme of fry fry [4].

It is interesting to observe how the energy received from food is distributed between body weight gain, sex glands, respiration and other expenses. Whitefish juveniles use food to the maximum: the ratio of the increase in ichthyomass to assimilated food sometimes reaches 60%; then drops to thirty-five percent by the end of the first year of their lives, and also to twenty percent by the third year. The percentage of digestible food in whitefish is no more than 80% of their diet. With such ratios of the expenditure of forces for the growth of biomass and respiration, fish can hope for the required size of the waist of food with allowable daily amounts of food. Theoretically, according to the model, by the age of 10, a whitefish is able to gain 6 kilos in weight. In fact, the maximum weight of a whitefish varies within a few kg, being limited by interspecific competition. Age groups: three-4 and 5, which together make up over 80% %% of the total population. Males mature at age 2 when they reach a weight of more than 480 g; females mature at age 3 when reaching a weight of 650-1000 g. The model assumes that fish win spawning when they reach a weight of 700 g. With an equal sex ratio in the spawning stock, we assume that reproduction is currently consumed by about 22.5% masses of spawning fish. Each presence in spawning ends with a decrease in survival. Usually 1-3 spawning fish die. A similar post-spawning death of whitefish was observed in nature [five or six, 9, 11].

Our identification method was used without paying for a step-by-step approach, which involved model decomposition, block system superstructure and subsequent accounting for internal relationships, and external influences in simulation experiments.

At 1 meter step, the submodels were tuned to the standard conditions.In the submodel of individual weight dynamics, the functions that determine the dependence of the amount of food and the loss for respiration on the weight of the fish are selected and corrected, based on information about the standard diet, the distribution of forces received from food and the growth rate. A good agreement between the actual and calculated data was obtained in the search for power-law dependences of the amount of food and breathing costs on a huge number of fish with an intensive distribution of the received heat for fish of 2-10 years of age. In the generation dynamics submodel, the value of the natural mortality rate (cm) in the stable population was determined based on the information on the age structure, the above assumptions on age-related changes in the cm, and the standard catch rate f = 0.45. The submodel of the dynamics of caviar output determined the dependence of the amount of spawned eggs on the number and years of fish, based on data on the fertility of whitefish in the lake. Sevan.

At the 2nd stage of identification, the submodels were combined and adjusted together. The effect of population size on fish loads was taken into account by introducing the actual diet, which depends on the nutritional needs of the fish and the availability of food. For the maximum allowable amount of food, such an amount was taken that allowed the whitefish to restore a mass of 6 kg by the age of 10 years. The natural mortality rate was decomposed into three components: normal mortality with standard nutrition and no spawning, reproductive mortality, and starvation mortality. Starvation mortality was determined using data on fluctuations in fish weight by age group. Assume that fish that do not reach the minimum weight die. Mortality was determined based on the ratio of the actual volume of food to the maximum. The threshold is equal to the ratio due to which the sig can be accelerated to only the minimum weight. It is known that mortality is higher in the case of food deficiency and excess density of juveniles when introduced into a reservoir [4], which was confirmed by our girl for sturgeon. On the normative amount of food, calculated from the average monthly temperature of vagueness and the plot in the arithmetic of oxygen, Percentage Calculator determined by the balance equation according to the method of g. G. Vinberg [12]. Some environmental

Conditions that are suitable for adult fish can be detrimental to caviar. The scheme took into account the effect of temperature, ph, and dissolved oxygen content on egg mortality rates. When calculating the natural mortality of mature fish, the boundary values of abiotic factors known for a given species were used.

Tuning the model showed that the introduction of only a few causal relationships made it possible to reflect the main ecological aspects of the whitefish population in its stable state . Food restrictions prevent the population from growing indefinitely. The effect of population density on growth rate allows the whitefish to gain weight; eating the wrong foods will slow the growth and maturation of the fish, in extreme cases this will lead to their death. In the scheme, fish can almost never ask for as much food as they want, so the actual amount of food consumed is on average 0.6 of the maximum number. The scheme successfully takes into account the known ecological facts: 1) the life span of an individual depends on the age of its first spawning; 2) whitefish don't mind skipping some spawning seasons; 3) spawning reduces the chances of survival.

Results of computational experiments

Standard environmental conditions are chosen for the pleasure of setting up the model, and conducting experiments. For the initial state of the model population, the abundance n of fish and telephone biomass b (by age groups) were taken from a population of 16.4 million fish with a total mass of 7124 tons according to 1980. After that, the age groups iteratively change their abundance/biomass. The greatest changes occur in the team of underyearlings, where the average weight increases by 5 orders of magnitude, and the number decreases by 3 orders of magnitude. With 6.64 billion larvae at the beginning, the total mortality of whitefish larvae and fry is 99.8%. At the end of the year, 7800 million individuals remain from any hatched larvae. High mortality cannot prevent a rapid increase in the biomass of the population. The rate of weight gain drops rapidly during frosts. The relative loss of body weight caused by spawning is constant in almost all age groups from three to ten centuries and is equal to 22.5%.

The average annual investment in respiration is 31% in the 1st age group and 68% in 2. In the older groups, it almost does not change, accounting for about 80% of the diet. The feed efficiency ratio (f/a) changes little with the number of years, ranging from about 19.4 to 19.8%.

An important design scenario was to consider when the whitefish population would respond to changes in fishery and food. Let's assume that intensive

Fishing occupies a significant part of the population, without selection for fish size. The number and biomass of the population are falling sharply, decreasing by about 100 times in ten years. Fishing undermines itself: by the year 20 it can catch only 0.7 tons with a population of about 1 thousand fish. In nature, such a reduction in population usually leads to irreversible degradation; this is what happened with canadian cod or caspian stellate sturgeon. A similar effect of overfishing was observed in lake sevan in the 2000s. But sometimes people, the peculiarities of the structure of the whitefish population (the formation of local subpopulation groups) provide them with a chance to restore their numbers based on partially preserved reproductively isolated groups. Whitefish population in the canadian part of the lake. Ontario successfully recovered its numbers in the 1990s [25] after being overfished in the 1960s. In our model, a strict ban on fishing from the age of 21 allowed the population to gradually resume its abundance. Calculations have shown that the population can be destroyed in 10 years, and repair without fishing would take more than 20 years.

Hunting intensity af = [0.30; 0.45] keeps the population at a height of 20-40 million fish with a biomass of twelve-18 thousand tons, maintaining stable catches at the level of 3000 tons. An increase in fishing intensity and a decrease in the minimum weight of the caught fish quickly deplete biological resources. Overfishing is reducing populations like never before with wild fluctuations in food supplies. The ecological niche is occupied by competing species of lower value. Recent work on other whitefish populations confirms findings about the effect of density factors on fish growth rate [26], the importance of eutrophication for the welfare of lake populations in canada and germany [27, 28], and the effect of temperature [27, 28]. 29].

1. Mikhailov v.V. Perevaryukha a.Yu. Modeling the dynamics of nutrient load in the course of assessing the effectiveness of replenishment of biological resources. Information and control systems. 4. P. 103–110. Doi: 10.15217/issn1684-8853.2017. 4.103

2. Solovieva tn, perevaryukha a. Yu. Dynamic model of depletion of sturgeon stocks with a complex intrapopulation structure. Information and control systems. 4, pp. 60-67. Doi:10.15217/issn1684-8853.2016.4.60

3. Reshetnikov yu. S. Tereshchenkov vg quantitative level of fish ecology research and error

Based on the data on the introduced species of the sevan whitefish, a segment model was developed that reproduces the dynamics of the number and biomass of fish. The age groups of the fish population are dependent on the fishery and environmental factors such as food availability, temperature, indistinct ph and dissolved oxygen concentration. The model can generally be categorized as simulation, deterministic, and discrete-continuous. The model has been identified and verified with the obligatory biological substantiation of its results by information on whitefish stocks [17, 23]. The model can work in two modes: under standard natural conditions (without specifying the hydrological situation) and with a scenario of anthropogenic changes. The scenario approach makes it possible to use data on lake eutrophication to predict the modification of the structure of fish citizens when abiotic factors change. As a result, the displacement of the upper level of the trophic chain will affect the entire biotic community of the lake. The scenarios demonstrate the inexpediency of breeding individuals that make the greatest contribution to the rate of biomass growth. The depleted population will recover at a rate lower than expected.

This study confirms the conclusion made in our previous rfbr projects about the efficiency of reproduction of the caspian sturgeon. Even small fluctuations in fish mortality in early ontogeny lead to changes in the population size, which is difficult to take into account in commercial forecasts based on the statistical averaging of a data array, at least a large one, but plotted in different situations of the existence of a population. And unstable hydrological situation caused by the needs of hydropower [30].

The study was supported by the rfbr grant 1707-00125.

Associated with it. Russian ecological journal, 2017, v. 1, p. 48, no. 3, pp. 233–239.

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