Math 225
Sect 001
Allan Meldrum
30204119
In analyzing growth, decay, and equilibrium states of a population-based system over time, a valuable modelling tool comes in the form of the ordinary differential function for logistic growth,
1 𝑑𝑃
𝑃 𝑑𝑡
= 𝜌(𝑘 − 𝑃) , (1) where P represents the current population or observed quantity of a system, k is the carrying capacity of the system, and 𝜌 is a scaling constant or representative of maximum growth rate [1]. According to this equation, a population of zero will not vary, as is true of a population which resides at the carrying capacity. Since 𝜌 is positive, the population decays when above the carrying capacity and grows when below the carrying capacity. Thus, there is an equilibrium point at the carrying capacity value. To solve equation (1), separation of variables is used such that
1 𝑑𝑃
𝑃 𝑑𝑡
= 𝜌(𝑘 − 𝑃) ⟺ ∫
1
𝑃(𝑘−𝑃)
= ∫ 𝜌𝑑𝑡 ⟺
𝑃 𝑘−𝑃
= 𝐶𝑒 𝑘𝜌𝑡 ⟺ 𝑷(𝒕) =
𝑪𝒌
𝑪+𝒆 −𝒌𝝆𝒕
, (2) where C is a constant of integration [2]. This model is applied to numerous fields of study, and below we will explore two examples.
One application of logistic growth is in the study of urban growth using fractal dimension evolution. According to Y. Chan [3], the normalized data obtained from the timedependent fractal dimension of urban form may be expressed as 𝑑𝐷(𝑡)
= 𝑘𝐷(𝑡)[1 − 𝐷(𝑡)], 𝑑𝑡
(3) where D(t) is the fractal dimension as a function of time and k is the initial growth (3).
Equation (3) shows that the equilibrium point is reached when D(t) = 1. Solved by Chen, the solution to (3) is
𝐷(𝑡) =
1+(
1
1
−1)𝑒 −𝑘𝑡
, (4)

Math 225
Sect 001
Allan Meldrum
30204119 where 𝐷 ∗
0
= (𝐷
0
− 𝐷 𝑚𝑖𝑛
)/(𝐷 𝑚𝑎𝑥
− 𝐷 𝑚𝑖𝑛
) , the normalized result of D
0
[3]. Equation (4) can predict the fractal dimension of a system’s urban form as a function of time, the initial growth rate, and a normalized initial fractal dimension D
0
. Equation (4) may be used to optimize the spatial efficiency of an urban population by predicting the required fractal dimensions of the population at any given time.
Another application of equation (1) is the analysis of the global economy and the competitiveness of nations. From W. Kwasnicki, logistic and exponential models may be used to predict the future state of the global economy based on past national GDPs [4]. The logistic equation proposed by Kwasnicki is 𝑑𝑦 𝑑𝑡
= 𝑟𝑦 (1 − 𝑦
𝐾
) , (5) where y is the economic development of a population, r is the maximum growth rate, and K is the saturation level. The solution to Kwasnicki’s equation is presented as 𝑦(𝑡) =
𝐾
1+𝑎𝑒 −𝑏𝑡
, (6) where a and b are integration parameters. Equations (5) and (6) share the usual logistic growth traits, demonstrating a state of equilibrium around the point y = K. It is thought that with this set of equations and previous years’ data, the global population’s economic quality may be predicted at any point in time.
Where logistic growth models are useful in many cases, exponential growth models also have benefits. The shape of an exponential growth plot does not have the equilibrium state associated with logistic growth, and thus cannot model populations which have growth restrictions, although this becomes useful when modeling infinite growth of a population without growth restrictions, such as a well-fed bacterial culture [5].
Realistically, logistic growth models are often used in this case as well, as no population will have an infinite amount of resources in the real physical world [6]. Logistic growth models provide a more realistic modelling of the physical world, finding use in virtually any field of study, and are thus an invaluable tool in mathematical analysis and modelling.

Math 225
Sect 001
References
Allan Meldrum
30204119
[1] R.K. Nagle, E.B. Saff, and A.D. Snider, editors. Fundamentals of Differential Equations and
Boundary Value Problems. Pearson, Boston, MA, 2005.
[2] B. Kooi, M. Boer, & S. Kooijman. On the Use of the Logistic Equation in Models of Food
Chains. Bulletin of Mathematical Biology, 60(2): 231–246. 1998.
[3] Y. Chen. Fractal dimension evolution and spatial replacement dynamics of urban growth. Chaos, Solitons & Fractals, 45(2): 115–124. 2012.
[4] W. Kwasnicki. Logistic growth of the global economy and competitiveness of nations.
Technological Forecasting & Social Change, 80(1): 50–76. 2013.
[5] Wikipedia.org. Exponential growth. http://en.wikipedia.org/wiki/Exponential_growth.
17 Jan. 2013.
[6] C. Goudar. Computer Programs for modeling mammalian cell batch and fed-batch cultures using logistic equations. Cytotechnology, 64(4): 465–475. 2012.