Mathematical modelling of biological populations with and without dispersion

Abstract

All the biological populations living in natural environment have attracted mathematicians and statisticians since the beginning of last century. A pragmatic study of such populations leads to mathematical modeling involving different species living under different situations and conditions, in such case the environmental and physical parameters play important role in generating appropriate models in terms of mathematical expressions including difference and differential equations, mostly non-linear in nature for different categories of animal and microbial populations have associated parameters of specific nature. In this paper I cover a variety of interacting and non-interacting living beings on the earth surface both infinite numbers with basic features like growth/ reproduction, decay/death and migration from and within one domain to another. This involves advanced mathematical tools to fit into the process for the study leading to pattern recognition and forecasting, impact of environment and the topography. I have incorporated effect of population dispersion in certain cases.

Keywords:

• Mathematical modeling
• biological populations
• dispersion

Introduction

Population dynamics is an important area of biomathematics and biostatistics. The population studies are associated with many processes in life sciences such as natural growth and culture of microorganisms, growth and decay of viruses, growth, decay and migration in aquatic and wild life and demography. These studies differ from one population species to other depending on the type of the growth and environment involved. Some populations are isolated and grow independently with each unit dividing itself or grow out of heterogeneous pair contributing to reproduction. While in other cases two or more species of population live together and may be dependent or independent on each other. In all cases the environment plays a key role along with other factors such as availability of nutrients, presence of toxicants and physic-chemical condition. Thus, population dynamics is a subject with many applications. It also plays role in other studies like epidemiology, pollution analysis and ecology. In particular, it plays a major role in the management and manipulation of protected wild life.

Any biological population has the following important factors:

• Growth/Reproduction

• Decay/Death

The Mathematical behavior has the following types with respect to time:

• Discrete

• Continuous

• Piece-wise continuous

• Transitional

The mathematical modeling of each may differ with others in several respects. This difference may be due to various factors like natality, mortality, fertility, community behavior, the impact of environment and genetic effect.

Populations are groups of individuals belonging to the same species that live in the same region at the same time. Populations, like individual organisms, have unique attributes such as:

1. Growth rate

2. Age structure

3. Sex ratio

4. Mortality rate

Populations change over time due to births, deaths, and dispersal of individuals between separate populations. When resources are plentiful and environmental conditions are appropriately favorable, populations can increase rapidly. A population’s ability to increase at its maximum rate under optimal conditions is called its biotic potential. Biotic potential is represented by the letter r when used in mathematical equations.

In most instances, resources are not unlimited and environmental conditions are not optimal. Climate, food, habitat, water availability, and other factors keep population growth in check due to environmental resistance. The environment can only support a limited number of individuals in a population before some resource runs out or limits the survival of those individuals. The number of individuals that a particular habitat or environment can support is referred to as the carrying capacity. Carrying capacity is represented by letter K when used in mathematical equations.

Populations can sometimes be categorized by their growth characteristics. Species whose populations increase until they reach the carrying capacity of their environment and then level off are referred to as K-selected species. Species whose populations increase rapidly, often exponentially, quickly filling available environments, are referred to as r-selected species.

Characteristics of K-selected species include:

• Late maturation

• Fewer, larger young

• Longer life spans

• More parental care

• Intense competition for resources

Characteristics of r-selected species include:

• Early maturation

• Numerous, smaller young

• Shorter life spans

• Less parental care

• Little competition for resources

Some environmental and biological factors can influence a population differently depending on its density. For example, if individuals are cramped in a small area, disease may spread faster than it would if population density were low. Factors that are affected by population density are referred to as density-dependent factors.

There are also density-independent factors which affect populations regardless of their density. Examples of density-independent factors might include a change in temperature such as an extraordinarily cold or dry winter.

Another limiting factor on populations is intraspecific competition which occurs when individuals within a population compete with one another to obtain the same resources. Sometimes intraspecific competition is direct, for example when two individuals vie for the same food, or indirect, for example when one individual’s action alters and possibly harms the environment of another individual.

Equations of growth and decay

There is one elementary principle which underlines all population modeling, a principle which is too easily forgotten when battling with complex mathematics, namely that total number of individuals (N) in a fixed region of space can only change for four reasons:

1. Births

2. Deaths

3. Immigration

4. Emigration

The growth and decay processes of interacting populations are not always continuous in limited or protected zones. Particularly the large and medium size animal population species may not change during small interval of time.

Single species large population (continuous growth)

Let $N(t)$ be the population of the species at time t, then the rate of change in the population is given by (1) $$\frac{dN}{dt}=\mathit{Births}-\mathit{Deaths}\pm \mathit{Migration}$$(1)

When there is no migration and the birth and death terms are proportional to N then (Murray 1993). (2) $$\frac{dN}{dt}=(b-d)N$$(2)

Where b and d are positive constants (per capita birth and death rates, respectively) in formulation of this model we assumed that resources like food, space, light etc., are not limited and are in abundance so that the growth of the population is not limited by the availability of resources. The initial population $N(0)$ is denoted by $N_0$.

Now solving the above equation, we get (3) $$N(t)=N_0{e}^{(b-d)t}$$(3)

If $b>d$ then population grows exponentially and if $b<d$ then it becomes extinct.

Interacting populations

These are the main types of interactions

1. Prey-Predator

2. Competitive

3. Mutualistic or Symbiotic

4. Mixed

Prey-predator models

For large populations we have that if $P(t)$ is prey population and $Q(t)$ predator population, then, (Volterra 1925; Volterra 1926). (4) $$\frac{dP}{dt}=P(a-bQ)$$(4) (5) $$\frac{dQ}{dt}=Q(cP-d)$$(5) $P(t)$ = prey population $Q(t)$ = predator populationa = growth rate of prey population (per capita)b = rate of predationc = prey dependent predator growth rated = death rate of predator population (per capita)

All a, b, c and d are positive constants.

The assumptions in this model are:

1. The prey in the absence of any predation grows unboundedly in a Malthusian way; this is indicated by aP term in equation.

2. The effect of predation is to reduce the prey per capita growth rate by a term proportional to prey and predator populations, this is represented by $(-\mathit{bPQ})$ term in equation.

3. The prey’s contribution to predator’s growth rate is cQP in equation; it is proportional to available prey as well as to the size of predation.

4. In absence of any prey for sustenance the predator’s death rate results in exponential decay.

Interacting population with competition

Two or more species compete for the same limited food source or in some way inhibit each other’s growth. A simple two species Lotka-Volterra competition model with each species P and Q having logistic growth in the absence of other is (Murray 1993) (6) $$\frac{dP}{dt}=r_{1}P(1-\frac{P}{K_1}-b_{12}\frac{Q}{K_1})$$(6) (7) $$\frac{dQ}{dt}=r_{2}Q(1-\frac{Q}{K_2}-b_{21}\frac{P}{K_2})$$(7) where $r_1$, $r_2$, $K_1$, $K_2$, $b_{12}$, and $b_{21}$ are all positive constants. The $r^{\prime }s$ are linear birth rates and $K^{\prime }s$ are the carrying capacities. The $b_{12}$ and $b_{21}$ measure competitive effect of Q on P and P on Q, respectively.

Mutualistic population groups

Mutualism is a positive reciprocal relationship between two individuals of different species which results in increased fitness for both parties. Mutualism may be symbiotic in which the organisms live together in close physical association. The simplest mutualism model equivalent to the classical Lotka-Volterra predator-prey model is (Murray 1993). (8) $$\frac{dP}{dt}=r_{1}P+a_{1}PQ$$(8) (9) $$\frac{dQ}{dt}=r_{2}Q+a_{2}QP$$(9) where $r_1$, $r_2$, $a_1$, $a_2$ are all positive constants. Since $$\frac{dP}{dt}>0 \text{and}\frac{dQ}{dt}>0,$$

P and Q simply grow uniformly.

Single species finite population

If a population P has finite number of individuals (possibility of both sexes and all normal age groups) is called a finite population. In this case P is discrete and is governed by difference equation (Gotelli 1995). (10) $$P_n=P_{n-1}+f(P_{n-1},\lambda )$$(10) where $P_n$ denotes population after n generations with $P_0$ as its initial value is an appropriate function depending on the type of species and environment, $\lambda$ is control parameter. If f is linear, say $f=\alpha P_{n-1}+\lambda$, then (11) $$\begin{array}{c}{P}_n={\alpha }^n{P}_0+\lambda (1+\alpha +{\alpha }^2+\cdots +{\alpha }^{n-1})\\ ={\alpha }^n{P}_0+\lambda \sum _{i=0}^{n-1}{\alpha }^i\end{array}$$(11)

Two interacting species with finite populations

Let $P_n$ and $Q_n$ are two interacting populations in nth generation. $\lambda$ and $\mu$ are the parameters then growth and decay processes of interacting populations are not always continuous in limited or protected zones. Particularly, large and medium size animal population species may not change during small interval of time. Not only the breeding season has to be taken into account but also the duration in which the population is bound to change.

For interacting species, we can write (12) $$\begin{array}{c}\Delta P_n=f(P_{n-1},Q_{n-1},n,\lambda )\\ \Delta Q_n=g(P_{n-1},Q_{n-1},n,\mu )\end{array}$$(12)

Age based population groups of finite size

In a prescribed domain of habitat, the size of population determines the approach for quantitative modeling. In many cases we have to consider a given population size as discrete number and its growth as a step function of time. In such situations the population size is relatively small and the growth rate is generally fixed during a given time interval. This interval we call a step or a generation. For all practical purposes we take this interval as a time unit. The population under consideration is self-contained in healthy environment and has enough amount of nutrition available. We assume that $P_n$ denotes the population size after nth steps or generations with initial population $P_0$.

Species dispersion

Species distribution is the manner in which a biological taxon is spatially arranged. Species distribution is not to be confused with dispersal, which is movement of individuals away from their area of origin or from centers of high population density. A similar concept is the species range. A species range is often represented with a species range map. Bio-geographers try to understand the factors determining a species’ distribution. The pattern of distribution is not permanent for each species. Distribution patterns can change seasonally, in response to availability of resources, and also depending on the scale at which they are viewed. Dispersion usually takes place at the time of reproduction. Populations within a species are translocate through many methods, including dispersal by people, wind, water and animals. People are one of the largest distributors due to the current trends in globalization and the expanse of the transportation industry. For example, large tankers often fill their ballasts with water at one port and empty them in another, causing a wider distribution of aquatic species.

Population change with diffusion

The biological and ecological consequences of pollution in our environment may be considered in several ways depending upon the toxic level of pollutants and the ecotoxicological situations. They have studied the effect of environmental pollution on the growth and existence of two interacting biological populations in the situation where the pollutant causes injury to the principal physiological and biochemical systems of the populations and their environment. To study this situation, a mathematical model is presented by them consider the growth of interacting and dispersing biological species of density $N_i(x,t),(i=1,2)$ in a one- dimensional linear habitat $0.5<x\le L$, whose growth rate and the carrying capacity of the environment is decreasing due to the environmental pollution present in the habitat. The dynamical equations governing the growth of the species are assumed to be given by the non-linear partial differential equations. (13) $$\frac{\partial N_i}{\partial t}=N_i{F}_i(N_1,N_2,r_i(C),K_i(C))+D_i\frac{{\partial }^2{N}_i}{\partial x^2},i=1,2$$(13) where, $F_i(N_1,N_2,r_i(C),K_i(C))$ determines the interaction function of the species. $r_i(C)$ and $K_i(C)$ are the intrinsic growth rate and the carrying capacity of the environment respectively which are affected by the concentration $C(x,t)$ of pollutant. $D_i>0,(i=1,2)$ is the dispersion coefficient of the species.

Song and Chen (2001) have studied conditions for global attractively on n patches predator-prey dispersion-delay models. In this paper a non-autonomous predator-prey dispersion model with functional response and continuous time delay is studied, where all parameters are time dependent.

Freedman, Shukla and Takeuchi (1989) have discussed effect of predator resource on a diffusion predator-prey system. In this work a model of a predator-prey system with diffusion and predator with resource is studied. In the research work of Misra and Jadon (1999) performed modeling in stability of system of two competing species with convective and dispersive migration in heterogeneous habitat. Searching predator and prey dominance in discrete predator-prey system with dispersion has been carried out by Yakubu (2008) the model for this study is an extension of a discrete time predator-prey model of Hassell (1978) that includes dispersion in a patchy habitat. In a single-patch system, it is shown that, irrespective of initial population densities, a prey species with a high growth rate dominates the system with resulting existence or extinction of the searching predator.

With a numerical example, it is shown that if searching predator preys exclusively on the prey species that has a high growth rate, then a spectrum of dynamical behavior including predator-induced coexistence, exclusionary dynamics, and Hopf bifurcation could occur in the single-patch system. If the searching predator preys exclusively on the prey species that has a low growth rate, then both the predator and its preferred prey species go extinct. An example is given illustrating that searching predator diffusion could change a system with stable coexistence of prey to extinction of prey species. In a two-patch system, it is shown that prey species with a high average growth rate dominates the system by driving all the prey species with low average growth rates to extinction. Dhar (2004) illustrated the situation of prey-predator system with diffusion and supplementary resources in a two patch environment.

Recently, many researchers including Vaishya (2007) extended simple linear models given by Saxena (1993) using Saxena’s I function by modeling of protected species biological systems. He has constructed single species discrete (protected) population models, both age structured and otherwise. The elementary interacting population models are available in the work of Caughley (1977) and Gotelli (1995).

Cushing et al. (2004) gave some discrete competition models and the competitive exclusion principle. A fundamental tenet in theoretical ecology is the “competitive exclusion principle”. According to this tenet, two similar species competing for a limited resource cannot coexist; one of the species will be driven to extinction.

This principle is supported by many mathematical models, the most famous of which is the Lotka-Volterra differential equation model for two competing species. It is well known that Lotka-Volterra model allows just four dynamic scenarios, all of which involve only equilibria as possible asymptotic states.

A coexistence case (in the form of a globally asymptotically stable equilibrium) occurs if competition between the species is weak. If, however, the inter-species competition is sufficiently strong, then competitive exclusion occurs (in the form of an equilibrium state possessing one zero component).

Discrete population models

Saxena (2011) considered confined population model of finite size in a prescribed domain of habitat and the size of population determines the approach for quantitative modeling. In many cases they have to consider a given population size as discrete number and its growth as a step function of time. In such situations population size is relatively small and growth rate is generally fixed during a given time interval. This interval we call a step or a generation. For all practical purposes we take this interval as a time unit. The population under consideration is self-contained in healthy environment and has enough amount of nutrition available. Suppose $P_n$ denotes the population size after nth steps or generations with initial population $P_0$.Then change of population in the nth step is given (14) $$\Delta P_n=P_n-P_{n-1}$$(14) which depends on various factors indicated above. Besides this, the population density or distribution, male–female ratio and the environment are also important.

In general, we can write $$\Delta P_n=f(P_{n-1},n,\lambda )$$ where f is some appropriate function assigned to specific species.

Here $\lambda$ is a free parameter accounting factors other than $P_{n-1}$ and n.

Broadly the change in population depends on the following three factors

• Birth rate (per capita) $B_n$

• Death rate (per capita) $D_n$

• Migration (immigration or emigration) $M_n$

Several migration patterns may be considered as follows:

1. Constant migration rate

2. Age-specific constant migration

3. Age-specific variable migration

We confine ourselves to linear form of function $f(P_{n-1},n,\lambda )$ and per-capita generation-based birth and death rates $B_n$ and $D_n$, i.e. $$\Delta P_n=B_n{f}_1(P_{n-1})-D_n{f}_2(P_{n-1})+M_n$$ or (15) $$P_n={\alpha }_n{P}_{n-1}+M_n$$(15) where $M_n$ is migration rate and ${\alpha }_n=1+B_n-B_n$. $f_1$ and $f_2$ are appropriate functions, if linear the solution of the difference equation can be given by (16) $$P_n=\prod _{i=1}^n{\alpha }_i{P}_0+\sum _{i=j}^{n-1}\prod _{i=j+1}^n{\alpha }_i{M}_j+M_n$$(16)

If birth and death rates are same throughout the generations, then $$P_n={\alpha }^n{P}_0+(M_n+\alpha M_{n-1}+{\alpha }^2{M}_{n-2}+{\alpha }^3{M}_{n-3}+\cdots +{\alpha }^{n-1}{M}_1)$$

Right hand side is a polynomial in $\alpha$.

Single species finite population growth with specified ratio in two age groups

We consider certain elementary models of finite population models of non-interacting species. This population is divided in two age groups, namely reproductive and non-reproductive. It is assumed that the two classes maintain a fixed ratio throughout the period under consideration. The length of each regular subinterval is called a generation and depends on particular animal or insect. This formulation is better than given by Saxena (2011) and extended form of a similar model introduced by Vaishya (2007).

Age based population $P_n$ may be divided into two parts, i.e. (17) $$P_n=P_n^{(1)}+P_n^{(2)}$$(17) where, $P_n^{(1)}$= Population of infants and juveniles (pre fertile)and $P_n^{(2)}$= Population in fertile age group

The change in population of first age group occurs due to births, deaths, migration and transition from one age group to second one.

Therefore, change in the population $P_n^{(1)}$ is given by (18) $$\begin{array}{c}\Delta P_n^{(1)}=P_n^{(1)}-P_{n-1}^{(1)}\\ \Delta P_n^{(1)}=B{P}_{n-1}^{(2)}-D_n^{(1)}{P}_{n-1}^{(1)}-T_n{P}_{n-1}^{(1)}+M_n^{(1)}\end{array}$$(18) where,B = Birth rate (uniform throughout). $D_n^{(1)}$ = Death rate in group one in $n^{th}$ generation. $T_n$ = Transition rate from $P_{n-1}^{(1)}$ to $P_{n-1}^{(2)}$ in nth generation. $M_n^{(1)}$ = Migration rate in one group in nth generation.

Hence, $$P_n^{(1)}=(1-D_n^{(1)}-T_n)P_{n-1}^{(1)}+B{P}_{n-1}^{(2)}+M_n^{(1)}$$

Similarly, $$\Delta P_n^{(2)}=P_n^{(2)}-P_{n-1}^{(2)}$$

So that (19) $$\Delta P_n^{(2)}=T_n{P}_{n-1}^{(1)}-D_n^{(2)}{P}_{n-1}^{(2)}+M_n^{(2)}$$(19) where, $D_n^{(2)}$ = Death rate in group two in nth generation. $T_n$ = Transition rate from $P_{n-1}^{(1)}$ to $P_{n-1}^{(2)}$ in nth generation. $M_n^{(2)}$ = Migration rate in second group in nth generation.

The solution of the difference equation can be given as $$P_n^{(2)}=\prod _{i=1}^n{y}_i{P}_0^{(2)}+\sum _{i=1}^{n-1}\prod _{r=i+1}^n{y}_r{M}_i^{(2)}+M_n^{(2)}$$

Replacing n by $n-1$, we get $$P_{n-1}^{(2)}=\prod _{i=1}^{n-1}{y}_i{P}_0^{(2)}+\sum _{i=1}^{n-2}\prod _{r=i+1}^{n-1}{y}_r{M}_i^{(2)}+M_{n-1}^{(2)}$$

Putting these values in equation, we get (20) $$P_n^{(1)}=x_n{P}_{n-1}^{(1)}+B[\prod _{i=1}^{n-1}{y}_i{P}_0^{(2)}+\sum _{i=1}^{n-2}\prod _{r=i+1}^{n-1}{y}_r{M}_i^{(2)}+M_{n-1}^{(2)}]+M_n^{(1)}$$(20)

The solution of difference equation is given by (21) $$\begin{array}{c}\begin{array}{cc}{P}_n^{(1)}\hfill & =\prod _{i=1}^n{x}_i{P}_0^{(1)}+B\sum _{r=1}^n\prod _{i=r+1}^n{x}_i\prod _{j=1}^{r-1}{y}_i{P}_0^{(2)}+\sum _{r=1}^n\prod _{i=r+1}^n{x}_i{M}_r^{(1)}\hfill \end{array}\\ +B\sum _{m=1}^{n-1}\sum _{r=m+1}^n\prod _{i=r+1}^n{x}_i\prod _{j=m+1}^{r-1}{y}_j{M}_m^{(2)}\end{array}$$(21) where, $$x_n=1-D_n^{(1)}-T_n$$ and $$y_n=1-D_n^{(2)}+T_n\left(P_n^{(1)}/P_n^{(2)}\right)$$

Pattern and growth of animal population with three age groups

Medium and large animal species with longer generation of life the population has to be divided in at least three groups (Sharma, Saxena, and Chaturvedi 2014); we can take (22) $$A_n=A_n^{(1)}+A_n^{(2)}+A_n^{(3)}$$(22) where $A_n^{(1)}$= Population of infants and juveniles (pre fertile) $A_n^{(2)}$= Adult population in the fertile age group $A_n^{(3)}$= Population of aged non-fertile

The change in population of first age group will take due to births, deaths and migration.

Therefore, the change in the population $A_n^{(1)}$ is given by the equation $$\Delta A_n^{(1)}=A_n^{(1)}-A_{n-1}^{(1)}$$

The governing equation can be written as (23) $$\Delta A_n^{(1)}=B_1{A}_{n-1}^{(2)}-D_1{A}_{n-1}^{(1)}-T_1{A}_{n-1}^{(1)}-q_{1}Q{A}_{n-1}^{(1)}+M_1$$(23) where $B_1$= Birth rate $D_1$= Death rate $T_1$= Transition rate from $A_{n-1}^{(1)}$ to $A_{n-1}^{(2)}$ $M_1$= Migration rateQ= Predator Population $q_1$= Predation rate in first group

Accordingly, we get the following difference equation $$A_n^{(1)}-A_{n-1}^{(1)}=B_1{A}_{n-1}^{(2)}-D_1{A}_{n-1}^{(1)}-T_1{A}_{n-1}^{(1)}-q_{1}Q{A}_{n-1}^{(1)}+M_1$$ or $$A_n^{(1)}=(1-D_1-T_{1}A-q_{1}Q)A_{n-1}^{(1)}+B_1{A}_{n-1}^{(2)}+M_1$$

Similarly, we have $$\Delta A_n^{(2)}=A_n^{(2)}-A_{n-1}^{(2)}$$

The governing difference equation is $$\Delta A_n^{(2)}=-D_2{A}_{n-1}^{(2)}-T_2{A}_{n-1}^{(2)}-q_{2}Q{A}_{n-1}^{(2)}+M_2+T_1{A}_{n-1}^{(1)}$$ or (24) $$A_n^{(2)}=(1-D_2-T_2-q_{2}Q)A_{n-1}^{(2)}+T_1{A}_{n-1}^{(1)}+M_2$$(24) where $D_2$ = Death rate $T_2$ = Transition rate from $A_{n-1}^{(2)}$ to $A_{n-1}^{(3)}$ $M_2$ = Migration rate $q_2$ = Predation rate in second groupSimilarly, we can write $$A_n^{(3)}=(1-D_3-q_{3}Q)A_{n-1}^{(3)}+T_2{A}_{n-1}^{(2)}+M_3$$ where, $D_3$ = Death rate $T_3$= Migration rate $q_3$=Predation rate in third group

As an illustration we compute values of $A_n^{(1)}$ for $n=7$, so that $$A_7^{(1)}=x{A}_6^{(1)}+B_1{A}_6^{(2)}+M_1$$ where, $$x=1-D_1-T_1-q_{1}Q$$ and the values of $A_6^{(1)}$ and $A_6^{(2)}$ in terms of previous values, we get (25) $$\begin{array}{cc}{A}_7^{(1)}=x^7{A}_0^{(1)}+B_1{T}_1{A}_0^{(1)}[\sum _{r=0}^4(5-r)x^{(5-r)}{y}^r\hfill & \hfill \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\sum _{r=0}^5{x}^r{y}^{5-r}+2x^2{y}^3]\hfill & \hfill \\ +B_1^2{T}_1^2{A}_0^{(1)}\left[4\sum _{r=0}^3{x}^{3-r}{y}^r+5\sum _{r=0}^2{x}^{3-r}{y}^r+x^3+3x^{2}y\right]\hfill & \hfill \\ +B_1^3{T}_1^3{A}_0^{(1)}(4x+3y)+B_1{A}_0^{(2)}\sum _{r=0}^6{x}^{6-r}{y}^r\hfill & \hfill \\ +B_1^2{T}_1{A}_0^{(2)}\left[5\sum _{r=0}^4{x}^{4-r}{y}^r+3\sum _{r=1}^3{x}^{3-r}{y}^r+x^2{y}^2\right]\hfill & \hfill \\ +B_1^3{T}_1^2{A}_0^{(2)}\left[6\sum _{r=0}^2{x}^{2-r}{y}^r+3xy\right]+M_1{p}_6(x)+B_1{T}_1{M}_1\hfill & \hfill \\ [\sum _{r=0}^4(r+1)x^r+2\sum _{r=1}^3{x}^{4-r}{y}^r+\sum _{r=1}^2{x}^{4-r}{y}^r\hfill & \hfill \\ +y(p_3(x)-1)+q_4(y)+2\sum _{r=1}^2{x}^{3-r}{y}^r]\hfill & \hfill \\ +B_1^2{T}_1^2{M}_1\left[\sum _{r=0}^2(r+1)y^r+3x+6x^2+6xy\right]\hfill & \hfill \\ +B_1{M}_2[p_5(x)+(p_3(x)-1)q_2(y)+y^3(p_2(x)-1)\hfill & \hfill \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+xy(x^3+y^3)+q_5(y)]\hfill & \hfill \\ +B_1^2{T}_1{M}_2[\sum _{r=0}^3(r+1)x^r+6x^{2}y+4xy+6x{y}^2\hfill & \hfill \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\sum _{r=1}^3(r+1)y^r]\hfill & \hfill \\ +B_1^3{T}_1^2{M}_2\left[1+3x+3y\right]+B_1^4{T}_1^3{A}_0^{(2)}+B_1^3{T}_1^3{M}_1\hfill & \hfill \end{array}$$(25) where, $$y=1-D_2-T_2-q_{2}Q$$ similarly from $$A_7^{(2)}=y{A}_6^{(2)}+T_1{A}_6^{(1)}+M_2$$

Putting the values of $A_6^{(1)}$ and $A_6^{(2)}$, we have (26) $$\begin{array}{c}{A}_7^{(3)}=z^7{A}_0^{(3)}+T_2{A}_0^{(2)}\sum _{r=0}^6{z}^{6-r}{y}^r+B_1{T}_1{T}_2{A}_0^{(2)}\\ [\sum _{r=0}^4{z}^{(4-r)}{x}^r+\sum _{r=0}^3(5-r)y^{4-r}{z}^r\\ +\mathit{xyz}(2z+3y+2x)+\sum _{r=0}^3(5-r)y^{(4-r)}{x}^r]\\ +B_1^2{T}_1^2{T}_2{A}_0^{(2)}[\sum _{r=0}^2{z}^{(2-r)}{y}^r\\ +x(2z+3y)+2\sum _{r=0}^1{y}^{(2-r)}{z}^r+3\sum _{r=0}^2{x}^r{y}^{(2-r)}]\\ +T_1{T}_2{A}_0^{(1)}[\sum _{r=0}^5{z}^{5-r}{x}^r\\ +\sum _{r=1}^5{z}^{5-r}{x}^r+x\sum _{r=0}^3{y}^{4-r}{z}^r+x^2\sum _{r=0}^2{y}^{3-r}{z}^r\\ +\sum _{r=1}^3{x}^{5-r}{y}^r+x^{2}y(y^2+xz)\\ +x^{2}y(y^2+xz)]+B_1{T}_1^2{T}_2{A}_0^{(1)}[\sum _{r=0}^3{z}^{3-r}{x}^r\\ +\sum _{r=1}^3{z}^{3-r}{y}^r+\sum _{r=1}^{3}r{z}^{3-r}{x}^r\\ +\sum _{r=1}^{3}r{z}^{3-r}{y}^r+4xyz+6\sum _{r=1}^2{x}^{3-r}{y}^r]\\ +M_3{r}_6(z)+B_1^2{T}_1^3{T}_2{A}_0^{(1)}(z+3x+3y)\\ +T_2{M}_2[r_2(z)-1+z{q}_4(y)+r_5(z)+q_5(y)+z^2{q}_2(y)\\ +z^3{q}_2(y)+y{z}^2(y^2+z^2)]\\ +T_1{T}_2{M}_1[z{p}_3(x)+z^2{p}_2(x)+z^3{p}_1(x)+p_4(x)+q_4(y)\\ +x{q}_3(y)+\sum _{r=0}^3{z}^{4-r}{y}^r\\ +(z+x^2)q_2(y)+\mathit{xyz}{p}_1(x)+y{z}^2{p}_1(x)+x{y}^{2}z]\\ +B_1{T}_1{T}_2{M}_2[r_2(z)+z^2{p}_1(x)\\ +p_3(x)-1+xz{p}_1(x)+2y{r}_2(z)+3y^2{r}_1(z)+2xy{r}_1(z)\\ +\sum _{r=1}^3(r+1)y^r{x}^{3-r}]\\ +B_1{T}_1^2{T}_2{M}_1[r_2(z)+2(x+y)r_1(z)\\ +3\sum _{r=0}^2{y}^{2-r}{x}_r+xy]+B_1^2{T}_1^2{T}_2{M}_2\\ \left[r_1(z)+2x+3y\right]+B_1^3{T}_1^3{T}_2{A}_0^{(2)}+B_1^2{T}_1^3{T}_2{M}_1\end{array}$$(26) where, $$z=1-D_3-q_{3}Q$$

As described by Saxena (2011) classical polynomials can be matched and compared with the polynomial solutions of finite animal populations. For non-age structured isolated species certain examples are given by him.

For two and more age groups of larger animal polynomials of two and three variables will play important role in the analysis and future projections of the growth and decay Extended Hermite polynomials of two variables in one such example which can be applied to the above population expressions.

Continuous population models

Population growth with diffusion of single species

If there is continuous dispersion /diffusion of the population in uni-directional medium with diffusivity D. Then the governing model obtained from Fick’s law is given below $$\frac{\partial P}{\partial t}=f(P)+\frac{\partial }{\partial x}\left(D\frac{\partial P}{\partial x}\right)$$

Population growth with diffusion of multiple species

The population growth with diffusion of multiple species is given by the equation $$\frac{\partial P_i}{\partial t}=P_i{F}_i(P_1,P_2,P_3,\dots ,P_n)+▽.(D▽P_i),i=1,2,3,\dots ,n.$$ where $▽$ is Laplacian in Cartesian or polar coordinates.

Population farming

Rasool et al. (2012) carried out theoretical analysis on the stability and persistence of interacting species during dispersion. In this paper one of the important phenomenon’s in the universe is the state of stability for every living organism. Investigation of stability and persistence of interacting species is one of the important aspects in theoretical population biology. The linear stability of prey predator system has been established by incorporating diffusion equation in order to estimate the dispersion among the species. They considered the growth of interacting and dispersive biological species of density $N_1(x,t)$ and $N_2(x,t)$ in a one dimensional linear habitat $0\le x\le 1$. The dynamical equations governing the following system of non-linear partial differential equations are $$\begin{array}{c}\frac{\partial N_1}{\partial t}=r_1{N}_1\left[1-\frac{N_1}{K_1}-{\propto }_1\frac{N_2}{K_1}\right]+\frac{\partial }{\partial x}(D_1▽N_1)\\ \frac{\partial N_2}{\partial t}=r_2{N}_2\left[-1+{\propto }_2\frac{N_1}{K_2}\right]+\frac{\partial }{\partial x}(D_2▽N_2)\end{array}$$ where $N_1$ = Prey population. $N_2$ = Predator population. $r_1$ = Intrinsic growth rate of prey population. $r_2$ = Intrinsic growth rate of predator population. $K_1$ = Carrying capacity of the prey population. $K_2$ = Carrying capacity of the predator population. ${\propto }_1$ = Predator response to the prey’s population. ${\propto }_2$ = Prey’s response to the predator’s population. $D_1$ = Dispersion coefficient of the prey population. $D_2$ = Dispersion coefficient of the predator population. $▽$ = Laplace operator.

Single species animal population with dispersion in two patches

As it is indicated that in large domains the biological populations disperse at micro-level. This process is same as diffusion of substances in a medium.

Here we consider a simple model of population change exponentially and its diffusion in linear direction. Such situations do occur in nature. Diffusion of fish population along with a water stream is one such example.

We determine the population concentration by using Finite Element Method. The mathematical equation is given by (27) $$\frac{\partial P}{\partial t}=\alpha P+\frac{\partial }{\partial x}\left(D\frac{\partial P}{\partial x}\right)$$(27) where $P(x,t)$ is population at time t and at a distance x from the origin. $\alpha =b-d$, (percapita birth rate-death rate) and D is diffusivity. The boundary and initial conditions are: $$\begin{array}{c}P(x,0)=0,\\ P(0,t)=P_0,\\ \frac{\partial P}{\partial x}=0,atx=l\end{array}$$ where l is the total length of the population habitat.

If we compare equation (27)(27) $$\frac{\partial P}{\partial t}=\alpha P+\frac{\partial }{\partial x}\left(D\frac{\partial P}{\partial x}\right)$$(27) with Euler-Lagrange equation $$\frac{\partial F}{\partial u}-\frac{d}{dx}\left(\frac{\partial F}{\partial u_x}\right)=0,>x\in [a,b],>u_x=\frac{\partial u}{\partial x},$$ which is equivalent to the variational integral (Myers 1971) $$I_{\mathit{opt}}={\int }_a^{b}F(u_x,u,x)dx$$

Then the variational integral of the problem is given by (28) $$I=\frac{1}{2}{\int }_0^l\left[D{\left(\frac{\partial P}{\partial x}\right)}^2-\alpha P^2+\frac{\partial }{\partial t}{P}^2\right]dx$$(28) for its optimum value, using Finite Element Method for the intervals $[0,{\alpha }_1]$ and $[{\alpha }_1,{\alpha }_2]$ for x, we take $$I=I_1+I_2$$

Where, (29) $$\begin{array}{c}{I}_1={\int }_0^{{\alpha }_1}\left[\frac{1}{2}{D}_1{\left\{{(P^{(1)})}^{\prime }\right\}}^2-\frac{1}{2}{\alpha }_1{(P^{(1)})}^2+\frac{1}{2}\frac{\partial }{\partial t}{(P^{(1)})}^2\right]dx\\ I_1={\int }_{{\alpha }_1}^{{\alpha }_2}\left[\frac{1}{2}{D}_2{\left\{{(P^{(2)})}^{\prime }\right\}}^2-\frac{1}{2}{\alpha }_2{(P^{(2)})}^2+\frac{1}{2}\frac{\partial }{\partial t}{(P^{(2)})}^2\right]dx\end{array}$$(29)

Two patches with respective population functions $P^{(1)}(x,t)$ and $P^{(1)}(x,t)$ are such that $$\begin{array}{c}P(x,t)=P^{(1)}(x,t),0\le x\le {\alpha }_1\\ P(x,t)=P^{(2)}(x,t),{\alpha }_1\le x\le {\alpha }_2\end{array}$$

We consider linear and quadratic approximations in the patches, so that $$P_1=A_1+B_{1}x$$

Further, we take $$P_2=A_2+B_{2}x+C_2{x}^2$$

Putting above values in equation (29)(29) $$\begin{array}{c}{I}_1={\int }_0^{{\alpha }_1}\left[\frac{1}{2}{D}_1{\left\{{(P^{(1)})}^{\prime }\right\}}^2-\frac{1}{2}{\alpha }_1{(P^{(1)})}^2+\frac{1}{2}\frac{\partial }{\partial t}{(P^{(1)})}^2\right]dx\\ I_1={\int }_{{\alpha }_1}^{{\alpha }_2}\left[\frac{1}{2}{D}_2{\left\{{(P^{(2)})}^{\prime }\right\}}^2-\frac{1}{2}{\alpha }_2{(P^{(2)})}^2+\frac{1}{2}\frac{\partial }{\partial t}{(P^{(2)})}^2\right]dx\end{array}$$(29) , evaluating and optimizing we arrive at algebraic equations. Solving the system and putting values in $P^{(1)}$ and $P^{(2)}$, we arrive at population patterns.

Predator based dispersion of prey population in areas of circular symmetry

There are natural habitats of wild animals where predator species play important role in distribution of prey population. Generally, these animals (prey) grow faster till they reach the carrying capacity and follow logistic growth pattern while the killer animals grow much slow.

In this topic estimation of prey population distribution is carried out in the presence of predators using variational Finite Element Method. The area under consideration is closure to circular symmetry.

The mathematical model of predator-based dispersion of prey population in areas of circular symmetry is given by (30) $$\frac{\partial N}{\partial t}=\alpha N-\beta PN+\frac{1}{r}\left[\frac{\partial }{\partial r}\left(Dr\frac{\partial N}{\partial r}\right)+r\frac{\partial }{\partial \theta }\left(\frac{D}{r}\frac{\partial N}{\partial \theta }\right)\right]$$(30) or (31) $$\frac{\partial }{\partial r}\left(Dr{N}^{\prime }\right)+r\frac{\partial }{\partial \theta }\left(\frac{D}{r}\frac{\partial N}{\partial \theta }\right)+\left(\alpha -\beta P\right)rN-r\frac{\partial N}{\partial t}=0$$(31)

Where, $$N^{\prime }=\frac{\partial N}{\partial r}$$ and $N(r,\theta ,t)$ = Prey PopulationP = Predator Population with negligible fluctuation $\alpha$ = Intrinsic growth rate $\beta$ = Predation rateD = Prey Diffusivity

Figure 1: Real map of animal habitat.

Boundary and initial conditions

$$\begin{array}{c}N(r,\theta ,0)=\eta (R_2-r)\\ N(R_0,\theta ,t)=N_0=e^{-\lambda t}(\alpha +\beta \sin \theta )\\ \frac{\partial N(R_2,\theta ,t)}{\partial r}=0\end{array}$$

The predator population P does not change significantly during the stipulated time while population along angular direction follows sinusoidal variation due to favorable topographical situation. Also the prey population does not cross outer boundary of zone.

Solution of the problem

Comparing the partial differential equation with Euler Lagrange equation (Myers 1971) we arrive at (32) $$\begin{array}{c}\begin{array}{cc}F\hfill & =\frac{r}{2}\left(Dr{N}^{\prime 2}\right)-\frac{r}{2}\frac{\partial }{\partial \theta }\left(\frac{D}{r}\frac{\partial }{\partial \theta }\right)N^2-\frac{r}{2}\alpha N^2\hfill \end{array}\\ +\frac{r}{2}\beta P{N}^2+\frac{r}{2}\frac{\partial }{\partial t}{N}^2\end{array}$$(32) the variational integral I in the region of radius $R_0\le r\le R_2$ is (33) $$\begin{array}{cc}I={\int }_{R_0}^{R_2}[\frac{r}{2}\left(Dr{N}^{\prime 2}\right)-\frac{r}{2}\frac{\partial }{\partial \theta }\left(\frac{D}{r}\frac{\partial }{\partial \theta }\right)N^2-\frac{r}{2}\alpha N^2\hfill & \hfill \\ +\frac{r}{2}\beta P{N}^2+\frac{r}{2}\frac{\partial }{\partial t}{N}^2]dr\hfill & \hfill \end{array}$$(33)

Figure 2: Idealized map of animal habitat.

Its optimum value is equivalent to the governing equation. If the annular region is further divided into two annular areas of radii $R_1$ and $R_2$. Then we have $$I=I_1+I_2$$

We consider linear shape functions represented as (34) $$N^{(1)}=A_1+B_{1}r$$(34) where, $$N(r,\theta ,t)=N^{(1)}(r,\theta ,t)\mathit{for}{R}_0\le r\le R_1$$

Then, at $r=R_0$, $N^{(1)}=N_0$

Suppose (35) $$N^{(2)}=A_2+B_{2}r+C_2{r}^2$$(35) here, $$N(r,\theta ,t)=N^{(2)}(r,\theta ,t)\mathit{for}{R}_1\le r\le R_2$$

Then, at $r=R_1$, $N^{(2)}=N_1$

Therefore, (36) $$\begin{array}{cc}{I}_1={\int }_{R_0}^{R_1}[\frac{r}{2}{D}_1{\left({(N^{(1)})}^{\prime }\right)}^2-\frac{r}{2}\frac{\partial }{\partial \theta }\left(\frac{D_1}{r}\frac{\partial }{\partial \theta }\right){\left(N^{(1)}\right)}^2\hfill & \hfill \\ -\frac{r}{2}{\alpha }_1{\left(N^{(1)}\right)}^2+\frac{r}{2}{\beta }_{1}P{\left(N^{(1)}\right)}^2+\frac{r}{2}\frac{\partial }{\partial t}{\left(N^{(1)}\right)}^2]dr\hfill & \hfill \\ I_2={\int }_{R_1}^{R_2}[\frac{r}{2}{D}_2{\left({(N^{(2)})}^{\prime }\right)}^2-\frac{r}{2}\frac{\partial }{\partial \theta }\left(\frac{D_2}{r}\frac{\partial }{\partial \theta }\right){\left(N^{(2)}\right)}^2\hfill & \hfill \\ -\frac{r}{2}{\alpha }_2{\left(N^{(2)}\right)}^2+\frac{r}{2}{\beta }_{2}P{\left(N^{(2)}\right)}^2+\frac{r}{2}\frac{\partial }{\partial t}{\left(N^{(2)}\right)}^2]dr\hfill & \hfill \end{array}$$(36)

On integrating and putting the values from equations (34)(34) $$N^{(1)}=A_1+B_{1}r$$(34) and (35)(35) $$N^{(2)}=A_2+B_{2}r+C_2{r}^2$$(35) , for optimum values, we arrive at the system of linear algebraic equations. On solving and putting the values of co-efficient in equations (35)(35) $$N^{(2)}=A_2+B_{2}r+C_2{r}^2$$(35) and (36)(36) $$\begin{array}{cc}{I}_1={\int }_{R_0}^{R_1}[\frac{r}{2}{D}_1{\left({(N^{(1)})}^{\prime }\right)}^2-\frac{r}{2}\frac{\partial }{\partial \theta }\left(\frac{D_1}{r}\frac{\partial }{\partial \theta }\right){\left(N^{(1)}\right)}^2\hfill & \hfill \\ -\frac{r}{2}{\alpha }_1{\left(N^{(1)}\right)}^2+\frac{r}{2}{\beta }_{1}P{\left(N^{(1)}\right)}^2+\frac{r}{2}\frac{\partial }{\partial t}{\left(N^{(1)}\right)}^2]dr\hfill & \hfill \\ I_2={\int }_{R_1}^{R_2}[\frac{r}{2}{D}_2{\left({(N^{(2)})}^{\prime }\right)}^2-\frac{r}{2}\frac{\partial }{\partial \theta }\left(\frac{D_2}{r}\frac{\partial }{\partial \theta }\right){\left(N^{(2)}\right)}^2\hfill & \hfill \\ -\frac{r}{2}{\alpha }_2{\left(N^{(2)}\right)}^2+\frac{r}{2}{\beta }_{2}P{\left(N^{(2)}\right)}^2+\frac{r}{2}\frac{\partial }{\partial t}{\left(N^{(2)}\right)}^2]dr\hfill & \hfill \end{array}$$(36) , we arrive at population patterns in two annular regions.

Conclusion

The model introduced and discussed in this paper are applicable to surface wildlife where all prey and predator species grow or decay together with diffusion in a directed zone with two patches of equal size. The populations do not grow beyond respective carrying capacities and diffusion coefficients are different .This is in conformity with the landscape of most of the non–marshy sanctuaries which have long ranges of forests and grassy patches .In most cases the patterns of all the populations are similar which show steadiness in the beginning and then increase together .This model provides a very useful tool which can be applied in various situations and can also be validated with practical examples.

This paper also emphasizes quantification of wild life populations of finite size which are seldomly discussed earlier. Such situations are common in scattered clusters. The models shown here are leading to polynomial forms of results are easy to compute mathematically and statistically. These cases can also be extended to stochastic situations and can be supported by field data. Infact, a new area of finite population dynamics can be developed on the basis of above study.

Acknowledgments

The author is indebted to Dr. Padam Sharma for streamlining and rearranging the contents of the paper and also Mr. Anand Pawar for secretarial help.

Disclosure statement

No potential conflict of interest was reported by the author(s).