In this paper, the focus is on the generalised Fisher type equation arising in population dynamics. The analysis of the generalised Fisher equation has been carried out using Lie point symmetries (see e.g. [1] ) and construction of conservation laws see e.g. [2] ). These types of equations have appeared in many fields of study, for example, the reaction-diffusion equations arise in heat transfer problems [3] , biology [4] [5] , and transmission of nerve signals [6] . The reaction-diffusion equations such as the generalized Fisher equation describe how the concentra- tion of a substance is distributed in space changes, whereby the diffusion term causes the spread over the surface. Fisher [4] had used a nonlinear reaction-diffusion equation to model the population growth of mutant genes over a period of time. One can take Fisher’s equation [4] and with simple modifications, and can derive the Fitzhugh- Nagumo equation [7] . Moreover, one can make a modification to the Fitzhugh-Nagumo equation in order to obtain Huxley’s equation [4] .

A significant amount of work has been done in the process of studying the reaction-diffusion equations. In particular, from the classical Lie symmetry analysis point of view (see e.g. [8] ), and nonclassical symmetry techniques [7] [9] [10] . It turns out that reaction diffusion equations such as the generalised Fisher equation admit the genuine nonclassical symmetries if the source term is given by a cubic (see e.g. [7] [10] ). In a recent work [3] [11] , the authors assume a diffusivity which depends on space variable. In this case, the diffusivity may be given as a power law function of space variable for the given reaction-diffusion equation to admit non- classical symmetries.

This paper is arranged as follows. In Section 2, we provide the mathematical models for problems arising in population dynamics. In Section 3, we provide a brief account of the symmetry methods. In Sections 4 and 5, we provide the nonclassical and classical Lie point symmetry reductions, respectively. In Section 6, we briefly provide remarks on the conservation laws of the equation in question. The discussions and concluding remarks are given in Section 7.

For a diploid population having two available alleles at the locus in question (and), there are three possible genotypes;, and, where is the allele is under observation. Then, the following three equation describe the change in the genotype frequencies, and,

where are the reproductive success rate of genotype is the common death rate, is the total population density, , and u is the frequency of allele. The frequency of allele is given by. Note that u depends on t and x. The frequency of allele is

Differentiating Equation (2) with respect to t, then the three genotype equations (1) collapse into a single equation that describes the change in frequency of the new mutant gene

where and. Equation (3) is a reaction-diffusion-convection equation with cubic nonlinearities referred to here as a generalised Fisher equation. If, the total population density is

constant in space, Equation (3) reduces to the Fitzhugh-Nagumo equation. When deriving the models with the continuous method, there is an extra convective term due to the assumption that total population density is not uniform spatially.

If one is to consider the conditions that Fisher had examined, so that the allele in question is completely recessive. This implies that the genotypes and have the same phenotype, therefore they have the same reproductive success rate. Let the alleles represented by A and a be set as the alleles and respectively. Hence, is the allele under consideration. If the one obtains

where. One can see that Equation (4) is also a reaction-diffusion-convection equation. If, then Equation (4) reduces to a generalized Fisher type equation,

which models the propagation of impulses along nerve axons. In this paper we focus on the reaction diffusion equation of the form

We refer to Equation (6) as the governing equation. The variable u may be viewed as representing population. The diffusivity may be given by quadratic in x for equations such as Equation (6) to admit genuine nonclassical symmetries (see e.g. [3] [11] ). In this paper the coefficient of the cubic source term is given by the power law where n is a real constant.

In this section, we restrict discussions to symmetry analysis of second order differential equations. Given for example, a second order differential equation of the form

where the subscripts denote all possible first and second derivatives of u with respect to t and x. Finding classical Lie point symmetries of Equation (7) implies seeking infinitesimal transformations of the form

generated by the vector field

Note that the transformations in (8) are equivalent to the one-parameter Lie group of transformations that leaves the Equation (7) unchanged or invariant. The action of X is extended to all derivatives appearing in the equation in question through the appropriate prolongation. The infinitesimal criterion for invariance of the given equation is given by

Equation (10) yields an overdetermined system of linear homogeneous equation which can be solved algorithmically. Note that the solution of Equation (10) yield the classical Lie point symmetries admitted by Equation (7). Full theory of determination of Lie point symmetries may be obtain in among other texts [12] - [14] . If the invariance is sought subject to a further constraint

known as the invariant surface condition (ISC), that is given

then one obtain a system of nonlinear determining equations which may yield the nonclassical symmetry generators [15] .

In this section, we consider nonclassical symmetry reductions of the generalised Fisher type equation given in Equation (6). Here, is given as a cubic function of u (see e.g. [7] ) and both the diffusivity and the coefficient of the source term are given by power law functions of the space variable.

It is worth noting that Equation (25) has the conservation laws given by the conserved vectors

provided which implies a linear heat equation with spatial dependent diffusion term. These combination of conserved vectors is obtained by both direct and multiplier methods.

In this paper, we have used both classical and nonclassical symmetry methods to construct the exact solutions. Some new group-invariant (exact) solutions for reaction diffusion equation with spatially dependent diffusivity and the coefficient of the source term have been constructed using both classical and nonclassical symmetry techniques. Figures 1-6 depict the change in mutant population with respect to either time or space or both. The effects of the parameters appearing in the plotted exact solutions on the population are displayed. We have introduced the modified Hopf-Cole transformation to transform a nonlinear second order ODE to a simpler to solve linear third order ODE. To the best of our knowledge, this transformation has never been used in the recorded literature.

KirstenLouw,Raseelo J.Moitsheki, (2015) Group-Invariant Solutions for the Generalised Fisher Type Equation. Natural Science,07,613-624. doi: 10.4236/ns.2015.713061