The Verhulst Model, alternatively known as the Logistic model, is a model that was developed by Belgian mathematician, Pierre Verhulst in the year 1938. It is a differential equation that relates the change in population size over time to birth and deaths that occur over time. This model is used on a population that has sufficient food supply and no threat from predators. This model does not assume unlimited resources. A population with this trait generally grows at a rate that is proportional to the population. This means that in each unit of time, a certain percentage of the citizens who are a part of that population produces a new group of individual. Unfortunately, most populations are constrained by limitations on resources. Logistic growth of a population generally begins with exponential growth at the initial part, due to the fact that the resources are plenty and few individuals to use them. As the population begins to grow further, due to the lack of unlimited resources, they start getting used up and hence, the population size increases at a slowing rate. Eventually, the rate of growth start to approach a plateau, which results in the following diagram. The model assumes there is the aforementioned carrying capacity, denoted by K, for the population. If the population is above K, then the population can be predicted to decrease. The steeper green curve represents an unconstrained pattern while the gentle blue curve refers to the population which is generally less than a number, denoted by K. When the size of the population is small as compared to K, the patterns are identical. The constraint of the environment does not cause a significant difference. With reference to the second population, as P becomes a significant fraction of K, the curves start to diverge and as P gets close to K, the rate of growth decreases to 0. Logistic growth can be modelled by mathematics using a modified equation for exponential growth. In this altered equation, the variable r, per capita growth rate, depends on population size, denoted by N, and is used to model how close it is to the carrying capacity. Carrying capacity refers to the maximum number of individuals of a given species which can be sustained by making use of the resources that are available in the said location. For humans, carrying capacity is a rather complex affair as compared to other organisms. This is due to the extension of the conditions required which help to define the carrying capacity. An example is, ensuring that cultural and social environments are not degraded and that physical environments are not adversely affected to the point where it may cause issues for the future generations. This is on top of the previously set definition which includes that the resources must not be significantly depleted or degraded because of the population living there. With regards to this particular model, it is assumed that the rate of change dNdt of the population size, N, and how far it is from converging the carrying capacity, K. dNdt= rmax(K-N/K)(N)At a particular point in time while the population is expanding, the “K-N” portion of the equation expresses the number of individuals that can still be added to the population before it exceeds carrying capacity. (K-N)/K refers to the fraction of the carrying capacity that has not been completely utilized yet. As a larger portion of carrying capacity has been used up, the value of (K-N)/K will reduce the growth rate. When the size of the population is rather small, the value of N is small compared to K. The (K-N)/K approaches 1, which then results in the exponential equation. Thus, this explains why the population growth is initially exponential but levels off more as it approaches K. The following equation can be obtained by manipulating the above to separate the variables. 1N(1-N/K)dN = r dtThis equation must then be integrated. While the right hand side of the equation may easily be integrated to become rt + C, the left hand side of the equation may only be integrated after its form has been changed into partial fractions, where it is expressed as the sum of two functions which are less complex. After substituting the value of the left hand side back into the equation, the manipulation of the equation by using algebra allows the final equation to be reached as such: , A is a constantA graph produced corresponding to this function has the asymptotes of y=K as well as the y-axis. It is also contains the point of inflection at y= K/2. To simply comprehend the equation, the value of 1 shall be substituted to each unknown value, which then gives rise to the following graph. It can be observed from the graph that the growth is nearly exponential. The graph then goes into saturation where it starts tapering towards a plateau and eventually converges with the carrying capacity, which in this case was 1. To provide a further example of how this model may be used, let there be a case in which unlimited resources are considered. This is only a hypothetical example as there is no real life example of a place where resources are unlimited. An assumption that is being made is that the value of the carrying capacity, denoted by K, is quite large compared to a typical population size. This results in =1/Kbecoming a rather small value due to the large denominator. The equation of the model is thus rewritten as ddtu=ru-ru2In order to determine the coefficient functions uj(t) for a range of 1 to N, an expansion of this equation is considered. u(t)=j=0Njuj(t), where N 2 In order to calculate the coefficient functions, the orders of are calculated where the terms with the same indec power as are collated together. The differential equations for the coefficient functions follow some initial conditions. The linear growth model is considered to be a leading order approximation to the Verhulst model and can be written as ddtu0=ru0The growth rate which declines to 0 can be accounted for by including a factor of (1-PK). The smaller the value of P, the closer the value of the factor is to 1, which means that there is no effect on the resulting value. With this, the resulting model is known as the Verhulst, or Logistic model.