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The effect of the alternative resource and time delay on conservation of forestry biomass is studied by considering a nonlinear mathematical model. In this paper, interaction between forestry biomass, industrialization pressure, toxicant pressure and technological effort is proposed and analysed. We find out the critical value of delay and observe that there is Hopf bifurcation. Using the normal form theory and the center manifold theorem, we determine the stability and direction of the bifurcating periodic solutions. Numerical simulations are given to illustrate the analytical results.

Forest is an integral part of our biosphere. It used for fuel, furniture etc. and thus provides strong foundation for the development of any country. Forest assists in the global cycling of water, oxygen, carbon and nitrogen. In many developing countries, people burn wood to get energy for heating and cooking. Forest also provides food and shelter to many wild life species. Due to overpopulation, industrialization and associated pollution forests are depleted alarmingly. A typical example is the Doon Valley in the northern part of India where the forestry resources are being depleted by limestone quarries, wood and paper based industries, growth of human and livestock populations, expansion of forest land for agriculture and settlement etc., threatening the ecological stability of the entire region [

1) To overcome the worldwide problem of conservation of forestry resources, synthetic is a good alternative of wood based product as it is cheap, and needs not much maintenance, and the one most important thing is that it looks fresher than wood based products.

2) Plastic and wood composite lumber are quickly becoming a common replacement for redwood, cedar, and treated lumber in such applications as decking, door and window frames, and exterior moldings. Redwood and cedar decking use virgin trees, maintaining our dependence on scarce wood resources. Plastic and wood composite lumber are worked similarly to real wood and do not require treatment, yet they hold up well to water, sun, insects, and salt air, typical enemies of wood [

[

In same year, [

As a consequence, we propose a model for the interaction of forestry biomass with industrialization pressure, toxicant pressure and applied technological effort. Further, the effect of alternative resource on the growth of forestry biomass is seen. The time delay is the inherent property of the dynamical systems and plays an important role in almost all branches of science and particularly in the biological sciences. In the further study of the model, we see the effect of time delay on the growth rate of forestry biomass. The rest of this paper is organized as follows: In Section 2, we analyze our model with regard to equilibria and their positive conditions. In Section 3, we investigate the stability of positive equilibrium and stability and direction of Hopf bifurcation. In Section 4, some numerical supports are carried out to justify the analytic results obtained in the manuscript. Section 5 deals with the conclusions of the paper.

We consider the following system of differential equations:

where

In model system (1),

where

Here

Here

Where

The system (3) is further modified when the technological effort

where

Here

Lemma: The region of attraction for the model system (4) is given by the set:

where

Equilibrium analysis: It can be checked that system (4) has four nonnegative equilibria namely,

Existence of

Here

From Equation (7), we get

Thus,

From Equation (6), we get

Putting the value of

Thus,

Existence of

Here

From Equation (10), we get

From Equation (8), we get

Putting the value of

Existence of

Here

From Equation (14), we get

After simple manipulation, we get from Equation (12) is

Putting the value of

Putting the value of

where

We note that

Remark 1. From Equation (15), it is easy to note that

brium density of forestry biomass increases as the growth rate coefficient of technological efforts and value of alternative resources increases.

To discuss the local stability of system (4), we compute the variational matrix of system (4). The entries of general variational matrix are given by differentiating the right side of system (4) with respect to

where

The variational matrix

The eigenvalues of matrix

The variational matrix

where

The characteristic equation corresponding to the variational matrix

where

According to Routh-Hurwitz criterion, equilibrium point

The variational matrix

where

The variational matrix _{22}, b_{23} and b_{44} are negative so the stability of equilibrium point E_{2} depends on sign of b_{11}. The equilibrium point E_{2} is stable manifold in

The variational matrix

where

The characteristic equation corresponding to the variational matrix

where

According to Routh-Hurwitz criterion, equilibrium point

To discuss the stability behavior of equilibrium

where

The linearized system of system (4) about

where

and

The characteristic equation for linearized system (19) is obtained as:

where

Let

Squaring and adding Equations (21) and (22), we get

where

Substituting

We assume that:

(H_{1}):

We notice that _{1}) holds and

Again solving (21) and (22), we get a critical value of delay given as follows

To investigate the behavior of the system (4) in the neighborhood of

Theorem:

We observe that the conditions for Hopf bifurcation are satisfied yielding the required periodic solution, that is,

This signifies that there exists at least one eigenvalue with positive real part for

Proof:

Differentiating Equation (20) with respect to

Therefore

We can obtain here

Verifying numerically it has been obtained that the transversality condition holds and hence Hopf bifurcation occurs at

In this section, we will derive explicit formulae for determining the direction, stability and period of the bifurcating periodic solutions arises through Hopf bifurcation. The method we will follow is based on the normal form theory and center manifold theorem as given in [

where

and

where

For

By the Reisz representation theorem there exists a function

In view of Equation (29) we can choose

where

and

The system (28) is the equivalent to

where

For

and a bilinear inner product

where

Suppose

which for

Solving the system of Equation (39), we get

and

Similarly calculating

where

Now the normalization condition gives

Thus,

Proceeding same as [

On the center manifold

We rewrite this equation as

where

It follows from (42) and (44) that

Also we have

where

So that

Thus

Now

Comparing the coefficients in (37) with those in (50), we get

In order to compute

with

Also, on

It follows from (46), (52) and (54)

etc. Now for

which on comparing the coefficients with (53) gives

From (56), (58) and the definition of

Note that

Similarly from (56), (59) and the definition of

where

It follows from the definition of

From Equations (61) and (63) we get

and

Using (61) and (66) in (64) and noting that

i.e.

Similarly using (63) and (67) in (65), we get

We solve system (69) for

Hence, using the results of [

Theorem (3.2.1): If

ing periodic solutions exist for

In this section, we present numerical simulation to illustrate the results obtained in the previous sections. The system (4) is solved using the MATLAB software package under the following set of parameters.

(a)

The interior equilibrium point of system (4) with data (a) is

Then, we can easily obtain that (H_{1}) to be satisfied. By computation, we have

The stability behavior of the system (4) for

refer

Now to verify the result of Theorem (3.2.1), we have shown the variation of variables

that the bifurcating periodic solutions arising from

In this paper, a nonlinear mathematical model is proposed and analyzed to see the effect of alternative resource and time delay on conservation of forestry biomass. We have obtained the explicit formulae that determine the stability and direction of the bifurcating periodic solutions by using the normal form theory and the center manifold theorem. For the given set of parameter values in (a), we found that, the Hopf bifurcation was supercritical with stable periodic solutions and the direction of bifurcation was

wood based industries by human awareness or some government action. Hence, we conclude from our analysis that the forestry biomass may be conserved by applying technological effort and alternative resources.

Second author thankfully acknowledges the NBHM (2/40(29)/2014/R&D-11/14138) for the financial assistance in the form of PDF.