E. Lindner et al. (eds.)Mathematical Modelling in Real Life ProblemsMathematics in Industry33https://doi.org/10.1007/978-3-030-50388-8_9

9. Mathematical Model for the Game Management Plan

Milana Pavić-Čolić¹

(1)

Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia

Milana Pavić-Čolić

Email: milana.pavic@dmi.uns.ac.rs

Keywords

Game management planRed deer managementPopulation dynamics

9.1 Introduction

We consider an enclosed hunting area, where the wild animals are grown, such as red deer, wild boar, mouflon. In those areas, humans can have a great influence on breeding. Usually, there is a management team that supervises animal raising, and thus an enclosed hunting area can be understood as a farm.

The overall goals of the management are to protect, sustain, and manage hunted wildlife, provide hunting opportunity, protect and enhance wildlife habitat, and minimize adverse impacts to residents, other wildlife, and the environment [1].

In this work we are particularly interested in the management activities of game population control. Usually this activity is focused on producing high quality trophies: the management controls a number of individuals over some period (or population dynamics) in order to raise the ones with trophy potential. The trophy is usually part a of the animal that the hunter keeps as a souvenir to represent the success of the hunt. It can be horn of a red deer or mouflon, teeth of a wild boar.

Trophy production is usually achieved by following two simple rules: (1) governing population of any initial structure towards the prescribed optimal one and (2) keeping high the number of trophy candidates. The population dynamics that obeys these rules is guided by the so-called Game Management Plan. It is a 10 years time frame document that in particular contains a plan of harvest for each hunting season, which is a main mechanism of population dynamics control.

In our study we focus on red deer population and the hunting area “Vorovo” in Serbia, whose management kindly provided the necessary data. In this case, trophy candidates are male deers 8–10 years old. All the predictions that are made in the Game Management Plan for this hunting area are based on experience and do not rely on any mathematical model, so far. Our aim is to put the problem on mathematical ground and to build a model that will improve the current management strategy keeping its simplicity. This is the first important result in direction of the game population control in enclosed hunting area such as “Vorovo”.

From the mathematical point of view, a very general framework can be derived, that can be applied to any enclosed hunting area that breeds any type of animals. Therefore, the answer that mathematics can provide would be useful all over the world.

This project was originally presented under the title “Red deer import” at the ECMI Modelling Week 2016, that took place in Sofia, Bulgaria. We studied the mathematical model for the Game Management Plan, and this paper partially contains the observations and results we obtained at the time. Besides that, we tried to understand the inbreeding problem, that is known to be the main problem in enclosed hunting areas. After some time living in those areas, the animals start to mate within family members, which provokes malformations and diseases in the new generations, and has irreparable consequences on trophy production. In order to avoid inbreeding, a hunting area imports new animals. The import process requires good strategy. First, the right time should be determined, and then the question of a number and an optimal structure of imported population regarding both sex and age arise. These problems are still open.

The plan of this chapter is as follows. In Sect. 9.2 we will introduce “Vorovo” hunting area and present the current management strategy, as well as some of its shortcomings. Section 9.3 is devoted to the new model for the Game Management Plan which aims to improve the current model, keeping its simplicity. We illustrate our model on the example of “Vorovo” hunting area in Sect. 9.4.

9.2 “Vorovo” Hunting Area

In this work, we are concerned with hunting areas of red deer population. In particular, we study the hunting area “Vorovo”, that is supervised by the Fruška Gora National Park in Serbia. This project aims at improving its strategy for production of the red deer trophies.

9.2.1 The Current Management Strategy

Let us explain the current management strategy for this specific enclosed hunting area. First, an optimal structure of red deer population is determined. This structure is fixed for the hunting area over the years. It is determined on the basis of environmental conditions, that include food and water availability, vegetation, climate, peace requirements, pedological composition of the soil (Table 9.1).

Table 9.1

The optimal structure of red deer population in “Vorovo” hunting area

		Age of individuals
		0	1	2	3	4	5	6	7	8	9	10
Number of individuals	Male	15	10	8	8	8	8	7	6	5	0	0
	Female	15	10	8	8	8	8	7	6	5	0	0

The first goal of the management is to reach this optimal structure for any initial population state. Achievement of this goal is driven by a 10 years time-frame document, the so called Game Management Plan.

The Game Management Plan contains for each hunting year (hunting year is from March 1 year until the March next year), within the interval of 10 years, the following data:

1.
optimal structure of population,
2.
number of individuals on March 31,
3.
number of individuals on April 1 (this date can be understood as “dears birthday”),
4.
expected number of newborns,
5.
number of individuals before hunting,
6.
expected loss,
7.
expected harvest,
8.
number of individuals after hunting,
9.
comparison of the number of individuals after hunting with the optimal structure.

We should mention that all the numbers that appear in the Game Management Plan are actually predictions. The only real data, besides the optimal structure, is a number of individuals on March 31 the first hunting year. In order to be sure to follow the Game Management Plan, hunting area needs to organize additional activities each hunting year. Two main such activities are game counting on March (at the beginning of hunting season) and making an Annual Plan. Annual Plan is a document whose aim is to plan the harvest so that the number of individuals at the end of the hunting season matches with the corresponding one from the Game Management Plan. Therefore, the Annual Plan is a sort of a comparison between real data and the predicted one in the Game Management Plan.

All predictions presented in the Game Management Plan are based on experience and do not rely on any mathematical model so far. In particular, harvest is planned in such a way that it ensures two tasks at the same time: (1) reach the prescribed optimal number of individuals at some moment, and (2) produce hunting trophies. Candidates for hunting trophies are 8–10 years old male deers. It is remarkable that the Game Management Plan preserves equilibrium state. More precisely, once the optimal structure is reached, the population dynamics do not change anymore (for example, the population structure before hunting is the same for each of the following years). Such equilibrium state is represented in Table 9.2.

Table 9.2

The equilibrium state

		Age of individuals
Population dynamics	Sex	0	1	2	3	4	5	6	7	8	9	10	Sum
Optimal number	Male	15	10	8	8	8	8	7	6	5	0	0	75	150
	Female	15	10	8	8	8	8	7	6	5	0	0	75
Number on March 31	Male	15	10	8	8	8	8	7	6	5	0	0	75	150
	Female	15	10	8	8	8	8	7	6	5	0	0	75
Number on April 1	Male	xx	15	10	8	8	8	8	7	6	5	0	75	150
	Female	xx	15	10	8	8	8	8	7	6	5	0	75
Expected growth	Male	21	xx	xx	xx	xx	xx	xx	xx	xx	xx	xx	21	42
	Female	21	xx	xx	xx	xx	xx	xx	xx	xx	xx	xx	21
Number before hunting	Male	21	15	10	8	8	8	8	7	6	5	0	96	192
	Female	21	15	10	8	8	8	8	7	6	5	0	96
Expected loss	Male	2	1	0	0	0	0	1	0	0	1	0	5	10
	Female	2	1	0	0	0	0	1	0	0	1	0	5
Expected harvest	Male	4	4	2	0	0	0	0	1	1	4	0	16	32
	Female	4	4	2	0	0	0	0	1	1	4	0	16
Number after hunting	Male	15	10	8	8	8	8	7	6	5	0	0	75	150
	Female	15	10	8	8	8	8	7	6	5	0	0	75
Relation to optimal number	Male	0	0	0	0	0	0	0	0	0	0	0	0	0
	Female	0	0	0	0	0	0	0	0	0	0	0	0

As we have already mentioned, besides this 10 years time frame document, hunting area organizes a lot of activities during the hunting year.

At the beginning, these activities are related to the observation of the real situation. First, in March (when the vegetation is still low) game counting is organized, when the real data is collected. Then the management makes a first version of the Annual Plan in April, which mostly aims at planning the harvest so that at the end of the hunting year the number of species coincides with the number from the Game Management Plan. In April, calving of females starts. Until the mid June, game warden monitors the hunting area, and in particular pays attention on losses. In mid June, with counted losses, hunting area makes a final version of the Annual Plan with a definite harvest prediction.

Then, the hunting is organized. First, in June and July hunting area organizes hunting for selective culling of newborns. Then, the main event is trophy hunting in September.

For the Modelling Week, “Vorovo” provided necessary data, more precisely the two Game Management Plans, for the intervals 2009–2019 and 2015–2025, as well as Annual Plans for hunting years 2010/2011, 2011/2012, 2012/2013, 2013/2014 and 2014/2015.

9.2.1.1 Some Shortcomings of the Current Management Strategy

When analyzing the data obtained from the park “Vorovo”, it can be noticed few shortcomings of the current management strategy. First, it is supposed to become stable in a short time, or in other words, the equilibrium state is supposed to be reached shortly after the period starts. According to the real data we get from Annual Plans, this seems unrealistic, since even the Annual Plan as a sort of control of the Game Management Plan is not successful in reaching desired structure at the end of hunting year.

Besides that, the population dynamics of newborns was not accurately modelled. According to the current model, the number of newborns was predicted as 70% of the number of females 2 years old and older. From the data we obtained, it was clear that the number of newborns was actually much higher.

Our main contribution is at improving this current management strategy. In the rest of the paper we explain our new proposed model and compare it with the current one. We intent to make a more accurate model for prediction of number of newborns, and to build a new model for the Game Management Plan that is less ambitious in how fast the equilibrium should be reached (and therefore seems more realistic) and that offers a good chance for trophy hunting.

9.3 A New Proposed Model for the Management Strategy

The first problem that attracted our attention was how to improve the current management strategy, or to obtain a new model for Game Management Plan, such that it keeps the fundamental goals:

1.
it predicts well the number of newborns,
2.
it reaches the optimal structure within 10 years,
3.
it keeps high the number of males older than 8 years, which are candidates for hunting trophies.

In the following two sections we describe our proposed model.

9.3.1 A New Model for the Number of Newborns

According to the current strategy, the number of newborns in one hunting year is planned as 70% of the number of females 2 years old and older on the date March 31 that hunting year. If one wants to check this model, it is needed to deal with the real data. This information is not available for the newborns. What is known is the number of “0” years old deers the following hunting season. In the meantime, we had harvest and some loss. Assuming that the harvest is realized with 100%, then one strategy for estimating the real number of newborns in one hunting year is the following: count number of “0” years old deers the next year and add the expected number of hunted and lost newborns the current year.

If we try to compare the current model with the real data obtained in such a way, we obtain a lot of disagreements, that are illustrated in Table 9.3, the third and the first row, respectively. We can see that, for example, in 2011/2012 the expected number of newborns is 66 according to the current model. But on March 31 the following 2012/2013 it was found 81 individuals, and 29 individuals was planned to be hunted in 2011/2012 . Therefore, if 29 individuals was hunted, then actually the number of newborns in 2011/2012 was 110. Thus, there is a mistake of 44 newborns. The same error appears in all other years for which we have data.

Table 9.3

Modelling the number of newborns: estimated number of newborns according to different models over hunting seasons

	Hunting season
	2010/2011	2011/2012	2012/2013	2013/2014	2014/2015
Real data with 100% realized harvest	102	110	90	68	81
Real data with 70% realized harvest	91	101	82	60	71
Current model	78	66	73	68	66
New model	86	76	90	82	70

Having discovered an inaccuracy of the current model, let us try to fix it. Still, we want to keep two main properties of the current model: (1) we want to have the number of newborns modeled as a certain percentage of certain population on the current year i.e. on March 31, and (2) once the equilibrium is reached, we need to stay in that state.

The big error in expected and real (under the assumption of 100% realization of harvest) number of newborns led us to assume that the harvest is realized with 70% of success. These new estimated real data are shown in the second row of Table 9.3.

In order to keep the main ideas of the current model, we looked into relation of number of newborns (real, with weighted harvest by 0.7) with respect to many cases: number of females of 2 years old and older, the sum of females older than 2 years and males older than 6 years and the whole population. We obtained that the relation of the number of newborns with respect to the whole population on date March 31, had the least standard deviation with respect to the average that we approximate with 21∕150. Moreover, as we will see, this number preserves the equilibrium state.

With this conclusion, we build our model. Let Σ _j denote the sum of the whole population on March 31 of the hunting year j∕j + 1, j = 0, …, 9. Then our new proposed model predicts the number of newborns, males and females, in the hunting year j∕j + 1, denoted with N _j∕j+1, as follows

$\displaystyle \begin{aligned} N_{j/j+1} = 2 \frac{21}{150} \varSigma_j. \end{aligned}$

(9.1)

The results are shown in the last row of Table 9.3.

In Fig. 9.1 we plot the real data (assuming harvest is realized with 70% of success) and predictions according to the current model and our new proposed model.

../images/475466_1_En_9_Chapter/475466_1_En_9_Fig1_HTML.png — Fig. 9.1
Number of newborns, the real data versus the current and our new proposed model

Note that in the equilibrium state for some hunting season j∕j + 1 the whole population sum on March 31 is Σ _j = 150, and therefore the number of newborns is then

$\displaystyle \begin{aligned} N^{eq}_{j/j+1} = 42, \end{aligned}$

(9.2)

which coincides with expected growth in equilibrium state shown in Table 9.2.

9.3.2 A New Model for the Game Management Plan

The current schema for the Game Management Plans do not follow any mathematical model, so far. It is based solely on experience. Given the real data on March 31 the first year of the Game Management Plan, the number of newborns is predicted, as well as losses and harvest within the whole population. So we obtain the state of population after the hunting current hunting year, which is precisely the population state on the date March 31 the next year. Then again newborns, losses and harvest are predicted and we obtain the population structure after the hunting the following hunting year. This procedure is repeated for the whole period of 10 years. We observed that (1) the Annual Plans as a sort of control of the Game Management Plan show that the real state differs significantly from the predicted one, and (2) the equilibrium state is reached very fast, in 2–3 years, which seems unrealistic (according to the Annual Plans, the equilibrium state was actually never reached!).

We propose a new model for the Game Management Plan that drives population dynamics for the next 10 years. Our model is more relaxed regarding expected harvest, in the sense that it predicts less hunting activities (and therefore seems more realistic than the current one). Consequently, the population structure according to our model does not go to the equilibrium state very fast, but it achieves the original goal of the Gama Management Plan, that is, it reaches equilibrium state within 10 years.

Our model is based on what we call survival probabilities. Let denote with p _i;j∕j+1 the survival probability of a male deer population of age i before hunting on the hunting year j∕j + 1, where i ranges from 0 to 10. Moreover, let M _i;j∕j+1 be the number of male deers of age i before hunting on April 1 the hunting year j∕j + 1. With the survival probability we calculate the number of deers of age i + 1 before hunting on the following hunting year j + 1∕j + 2, as follows

$\displaystyle \begin{aligned} M_{i+1; {j+1}/{j+2}} = p_{i; j/{j+1}} M_{i; j/{j+1}}. \end{aligned}$

(9.3)

For the female deers consideration is similar. We do not make difference in survival probabilities between males and females, but only in their numbers. We denote with F _i;j∕j+1 the number of female deers of age i before hunting on the hunting year j∕j + 1. Their number the following year is modelled with

$\displaystyle \begin{aligned} F_{i+1; {j+1}/{j+2}} = p_{i; j/{j+1}} F_{i; j/{j+1}}. \end{aligned}$

(9.4)

The initial state is the number of individuals 1 year old and older on April 1 the first hunting season 0∕1, represented by the matrix

../images/475466_1_En_9_Chapter/475466_1_En_9_Equ5_HTML.png

(9.5)

We resume the idea of our population dynamics model: take the initial state matrix (for the season 0/1), calculate the number of newborns according to the model we presented in Sect. 9.3.1, model the survival probabilities p _i;0∕1, i = 0, …, 10, and then get the population (of 1 year old deers and older) prediction for April 1 the next season 1/2. We repeat the procedure: calculate the number of newborns (for the season 1/2), from the modelling of survival probabilities p _i;1∕2, i = 0, …, 10, obtain the population (of 1 year old deers and older) prediction for April 1 the following season 2/3. The aim is to repeat this procedure iteratively for 10 years. Therefore, the main aspect of our model is to get the survival probabilities.

In what follows, we are going to model the population survival probabilities. We make a difference between the survival probabilities for newborns (i = 0) and all the other deers (i = 1, …, 10). Let us model first the survival probability for the population of deers older than 1 year in Sect. 9.3.2.1. Then in Sect. 9.3.2.2 we present our model of survival probabilities for newborns. Finally, we are going to apply our model on a concrete example in Sect. 9.4. For this, we will use the population state from the Annual Plan 2015/2016 as an initial state

../images/475466_1_En_9_Chapter/475466_1_En_9_Equ6_HTML.png

(9.6)

This is precisely the first data in the Game Management Plan 2015–2025 that we got from “Vorovo”, which allow us to compare our new proposed model and the current one.

9.3.2.1 The Survival Probabilities for 1 Year Old Deers and Older

When modelling the survival probabilities for the deers 1 year old and older, we take the simplest model. First, we assume all survival probabilities are constant over the years,

$\displaystyle \begin{aligned} p_{i; j/{j+1}} = p_i, \quad \forall j=1,\dots,9, \end{aligned}$

where i = 1, …, 10, and therefore our probabilities now depend only on deer age. In order to determine these probabilities, we recall that any model needs to preserve the equilibrium state once it is reached. Thus, we are motivated to look into the equilibrium state from Table 9.2 and extract information of the population state before hunting in two successive hunting years (Table 9.4).

Table 9.4

Determining the survival probabilities

../images/475466_1_En_9_Chapter/475466_1_En_9_Figa_HTML.png

We can see that the deer of age one (male or female) will “survive” and have 2 years the following hunting season with the probability $p_1 = \frac {10}{15}$ . Similarly, 2 years old deer will have 3 years the following hunting season with the probability $p_2 = \frac {8}{10}$ . In the same fashion we obtain all the other probabilities, represented in Table 9.5. We call them equilibrium survival probabilities since they are determined while referring to the equilibrium state.

Table 9.5

Equilibrium survival probabilities

$p_0^{eq}$	p ₁	p ₂	p ₃ = p ₄ = p ₅	p ₆	p ₇	p ₈	p ₉ = p ₁₀
$\frac {15}{21}$	$\frac {10}{15}$	$\frac {8}{10}$	1	$\frac {7}{8}$	$\frac {6}{7}$	$\frac {5}{6}$	0

For deers 1 year old and older for the survival probabilities we take the equilibrium ones. Only for newborns we make a different model in the following section.

9.3.2.2 The Survival Probability for the Newborns

In this section we model the probabilities p _0;j∕j+1, for j = 0, …, 9.

For the first hunting season 0∕1, we start with the given initial state (9.5). We calculate the sum of the whole population that year, denoted with Σ ₀,

$\displaystyle \begin{aligned} \varSigma_0 = \sum_{i=1}^{10} \left( M_{i; 0/1} + F_{i; 0/1} \right). \end{aligned}$

Then we predict the number of newborns according to the model (9.1),

$\displaystyle \begin{aligned} M_{0; 0/1} = F_{0; 0/1} = \frac{1}{2} N_{0/1} = \frac{21}{150} \varSigma_0. \end{aligned}$

(9.7)

These newborns (that count M _0;0∕1 or F _0;0∕1) will survive and will become 1 year old deers (males or females, respectively) the following hunting season 1∕2 with the probability

$\displaystyle \begin{aligned} p_{0; 0/1} = \frac{15}{M_{0; 0/1}} = \frac{15}{F_{0; 0/1}}. \end{aligned}$

(9.8)

Now the survival probabilities for the season 0∕1 are known: for p _0;0∕1 we take (9.8) and all the others p _i, i = 1, …, 10 are equilibrium ones from Table 9.5. They allow to predict the population state on April 1 the following season 1∕2, i.e. we can calculate M _i;1∕2 and F _i;1∕2, i = 1, …, 10 from (9.3) and (9.4), respectively.

For all the following hunting seasons j∕j + 1, j = 1, …, 9, the consideration is similar. Namely, from the previous step we first count the whole population on April 1 the hunting season j∕j + 1,

$\displaystyle \begin{aligned} \varSigma_j = \sum_{i=1}^{10} \left( M_{i; j/j+1} + F_{i; j/j+1} \right). \end{aligned}$

Then, following the model (9.1), we estimate the number of newborns that year

$\displaystyle \begin{aligned} M_{0; j/j+1} = F_{0; j/j+1} = \frac{1}{2} N_{j/j+1} = \frac{21}{150} \varSigma_j. \end{aligned}$

Finally, we model the survival probabilities of the newborns as

$\displaystyle \begin{aligned} p_{0; j/j+1} = \frac{15}{M_{0; j/j+1}} = \frac{15}{F_{0; j/j+1}}. \end{aligned}$

This probability in conjunction with p _i, i = 1, …, 10, from Table 9.5 gives M _i;j+1∕j+2 and F _i;j+1∕j+2, according to (9.3) and (9.4).

Note that the equilibrium survival probability for the newborns is

$\displaystyle \begin{aligned} p_0^{eq}= \frac{15}{21}, \end{aligned}$

(9.9)

using (9.2) and (9.7), which coincides with the equilibrium p ₀ from Table 9.5.

Taking the initial state (9.6), the newborns survival probabilities are plotted in Fig. 9.2, for the period 2015–2025.

../images/475466_1_En_9_Chapter/475466_1_En_9_Fig2_HTML.png — Fig. 9.2
The survival probability for the newborns by a hunting season. Dashed line represents the equilibrium probability $p_0^{eq}$

$$p_0^{eq}$$ — Fig. 9.2
The survival probability for the newborns by a hunting season. Dashed line represents the equilibrium probability $p_0^{eq}$

9.4 The Game Management Plan 2015–2025 According to the New Proposed Model

We present our new proposed model for the management strategy on an example. For the initial state we take the real data for the hunting season 2015/2016 from (9.6). Using survival probabilities for 1 year old deers and older from Table 9.5 and formulas (9.3) and (9.4), we get population state for 2 years old and older deer the following hunting season 2016/2017:

../images/475466_1_En_9_Chapter/475466_1_En_9_Equf_HTML.png

We calculate the sum of the whole population

$\displaystyle \begin{aligned} \varSigma_0= \sum_{i=1}^{10} M_{i; 2015/2016} + F_{i; 2015/2016} = 219.\end{aligned}$

Then we predict the number of newborns according to (9.1)

$\displaystyle \begin{aligned} N_{2015/2016} = 2 \frac{21}{150} \varSigma_0 = 62. \end{aligned}$

Therefore, from (9.7) we get the prediction of the number of newborns

$\displaystyle \begin{aligned} M_{0; 2015/2016} = F_{0; 2015/2016} = \frac{1}{2} N_{2015/2016} = 31. \end{aligned}$

Finally, the survival probability for the newborns is calculated from (9.8),

$\displaystyle \begin{aligned} p_{0; 2015/2016} = \frac{15}{31}. \end{aligned}$

Using (9.3) and (9.4) we obtain the deer population 1 year old for the following season 2016/2017:

$\displaystyle \begin{aligned} M_{1; 2016/2017} = 31 * \frac{15}{31} =15, \quad F_{1; 2016/2017} = 31 * \frac{15}{31} =15. \end{aligned}$

This completes the data for the hunting season 2016∕2017. Proceeding iteratively we build the Game Management Plan.

In Table 9.6 we present our results. For simplicity, we show only data for the number of the individuals before hunting for each year. The usual table (as the one for equilibrium, Table 9.2) used by “Vorovo” hunting area can be easily deduced.

Table 9.6

State of the population before hunting for the period 2015–2025. The initial state (9.6) and equilibrium state represented in Table 9.1 are framed

../images/475466_1_En_9_Chapter/475466_1_En_9_Tab6_HTML.png

From Table 9.6 it can be seen that we control the population mainly by controlling the lower diagonal part. We reach the equilibrium state by significantly reducing the population of newborns. This strategy has a great benefit—it gives the possibility of a good deer selection. When the optimal number of 1 year old deer is reached the second year (in this case 2016/2017) and successively for all the following years, it starts to propagate the equilibrium property, due to the equilibrium survival probabilities. From the other side, the upper diagonal part does not influence population dynamics (as the lower part does), so we leave high equilibrium probabilities. Leaving deers growing up in the upper diagonal part has a strong advantage—the number of candidates for the hunting trophies is high over the years.

In Fig. 9.3 we represent our model for the Game Management Plan for the period 2015–2025, by plotting the number of male deers and comparing it with their optimal number.

../images/475466_1_En_9_Chapter/475466_1_En_9_Fig3_HTML.png — Fig. 9.3
The number of male deers by their age according to the new proposed Game Management Plan ( ) versus the current one ( ) and their optimal number ( ), over the hunting seasons 2015–2025. Nb stands for newborns