# Simple Logistic Growth Model of Bold Ibis Population In Birecik, Turkey

Updated: Nov 24, 2022

Writers: Muhammed Emir Akat, Berçem Sevinç, Azra Baysal, Ahmet Kılıç

Diyarbakir Bahcesehir College Science and Technology High School. Sur, Diyarbakir. 21010

Dicle University Department of Biology. Sur, Diyarbakir. 21280

**ABSTRACT**

Bald ibis Geronticus eremita is a species found primarily on Arabian Peninsula. Due to human interference and extensive use of pesticides, the bald ibis population declined drastically. Throughout the 1950s to 2000s, the population failed to maintain its numbers. Currently, 285 bald ibises dwell in Birecik. This paper uses a logistic growth model on bald ibis data to calculate the approximate date of the population reaching the carrying capacity. The equation used 1988 and 2003 population numbers as a foundation. The carrying capacity will be reached after 140 years from the start. When 1988 is taken as the start point, it is estimated that 2130s is the expected date of the population reaching its carrying capacity.

**INTRODUCTION**

The bald ibis *Geronticus eremita *is an endangered species located as two ecologically and morphologically distinct, isolated groups in northwest Africa; Turkey and East Africa. The northwest African population breeds winters in Morocco and Algeria, while the other population breeds in Birecik/Turkey and East Africa. This paper will be on the Birecik population.

Five hundred pairs habited during the 1900s to the 1950s in Birecik, around 1000 to1300 birds were observed in 1911 by Weigold (1912-13) and in 1953 by Kumerloeve (1984). The colony diminished to 5 pairs in 186 and 7 pairs in 1987. Only 25 birds made it to 1988, making a decline rate of around 99% in 30 to 35 years. The main reason for the decline since the 1950s is the excessive use of insecticides by the Ministry of Agriculture and Forestry and health. From 1957 to 1958, a minimum of 200 bald ibises was found perished. Upcoming two years, 70 percent of Birecik’s population (600-700) died with the cause of acute insecticide poisoning (Hirsch, 1979). The use of the insecticide affected the bald ibis population for the coming 15 years. From 1960 to 1972, almost no young were produced. The fertile population diminished to 22 pairs by the end of 1973. (Hirsch, 1978) With the help of WWF and other voluntary organizations, the bald ibis population went under protection. Attempts to increase the population were made, but significant changes were yet to come. The notable increase in the population started around 1988, with ten fledglings, 15 fertile birds. In 2000, 45 birds were recorded. Until 2008, the rate of growth was inconsistent due to death and missing cases within the population. In 2009, 101 birds were recorded by the breeding station. (Akakaya, 1988)

Modeling the growth in a population has been executed in various methods for numbers of different conditions and densities. However, using a logistic growth model to predict the future population is the most appropriate within ecological sciences considering the laws of biology.

In this paper, we use a simplified logistic growth model and explain its parts. Firstly, we review each property of the equation. Then, we constitute our graphic with DESMOS.

**METHODOLOGY **

To have a simplified logistic model, we need to keep in mind what shapes a population. There are deaths and births at a rate in a population. Individuals will fight over resources and habitats. In a logistic model, the population, in the beginning, will grow increasingly. However, as time passes, the growth will slow down to a point and at last reach the carrying capacity.

Primarily, the data of bald ibis’s lifespan, gestation period, and initial population were gathered. Birth and death rates were reached depending on adult and juvenile bald ibis numbers. With these parameters, an exponential growth graphic had been drawn. However, a J-type graphic is obtained due to consideration of carrying capacity, resources, and competition. With the missing factors, a logistic growth graphic was drawn. The logistic growth formula was acquired, a variety of equations were used. An S-type graphic, which shows the time to reach carrying capacity, was obtained depending on population change by time. The stages of obtaining the formulas that used in drawing the logistic growth graphic were indicated below:

*1) dy/dt= Ky[1-(y/L)]*

In this stage, when *y *gets closer to zero if the population size is small, *y* is small, then *y* over *L* is tiny compared to one, soo this grows more or less exponentially. Nevertheless, if *y* is substantial, it is close to *L*, then one minus *y* over *L* will be very close to zero. So that will give us that it basically stabilizes, and it does not change anymore once it reaches *L*.

1) For procuring a general solution using a logistic growth formula, both sides of the equations were multiplied by* dt* and divided by *y[1- (y/L)].* Afterward, antiderivatives of both sides were found. Therefore, to procure a general solution using the logistic growth formula, both sides of the equations were multiplied by *dt *and divided by *y[1- (y/L)].*

* dt(dy/dt)= Ky [1- (y/L)]dt*

* dy/ y[1- (y/L)]= Ky [1- (y/L)]dt/ [1- (y/L)]*

* ∫ 1/ y[1-(y/L)]dy = ∫ K×dt*

2) The left side of the equation is divided and multiplied by *L*. *L* is distributed in the denominator by using the distributive property of the multiplier over the subtraction.

*L×1/ Y[1- (y/L)] ×L= L/ y(L-y)*

3) The equation found in the previous stage was equated to another equation consisting of *A* and *B* variables. For finding the *A* and *B* variables, both sides were multiplied by *y (L-y)*.

* y (L-y)[L/y(L-y)]= [A/y+ B/(L-y)] y (L-y)*

* L=A(L-y)+By*

* A=B=1*

Consequently, the equation found in the 2nd stage would be the equation below:

* ∫ 1/ y[1-(y/L)]dy = ∫ K×dt*

* ∫ [1/y + 1/(L-y)]dy= ∫ K×dt*

4) The general solution was obtained by finding the antiderivative of both sides. General Solution:

* P(t) = y= L/(1+be-Kt) *

The data of bald ibis were used in the general solution equation instead of the variables.

*L= Carrying Capacity= 1000*

*for t=0:*

*P(t) = initial population = 25*

*b= 39 *

*for t = 15:*

*P(t)= 63 *

*K= 0.06426 *

The logistic growth graphic was drawn by gathering the value of these variables.

**DISCUSSION**

In the 1930s, the recorded bald carrying capacity of Birecik was approximately 3000 birds. Due to the extensive use of DDT (dichloro-diphenyl-trichloroethane), the population declined from 1200 in the 1950s to 260 in 1962. Due to the usage of biocides, Birecik went through a substantial amount of habitat loss. As a result, their population decreased to 52 in 1973. The population failed to increase naturally because of electric wires, scarcity of food, and forced interbreeding. Therefore, the bald ibis population was taken into captivity to ensure the species’ safety. Due to the factors mentioned above, the carrying capacity will not likely reach back to its previous number, 3000. (1)

* (1) The authors would like to thank Amanda D. Rodewald from Cornell University for her recommendations.*

By precise calculations, it is estimated that the bald ibis population will reach its carrying capacity 140 years after the starting point of the equation, which is 1988. One needs to keep in mind that this is a very simplified model of logistic growth, which means that external factors such as disturbance of natural life, poaching, stress caused by global warming, the introduction of new members from other habitats, and juvenile data which tells that new members of the population cannot reproduce right are not taken into consideration. To take further steps on this research topic, one could execute a logistic growth model, taking the factors mentioned above into consideration. Including those factors would narrow down the proximity of the year required to reach the carrying capacity. This research shows us that humanity’s short-term actions have effects lasting for centuries.

**ACKNOWLEDGEMENTS**

The authors thank Professor Dr. Yüksek Coskun (Dicle University) for his recommendations, thank Professor Dr. Ahmet Kılıç (Dicle University) for his assistance during data collection, and thank Professor Amanda Rodewald for her suggestions for the improvement of the paper.

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