Mathematical Model of Tip-hypha Anastomosis and Dichotomous Branching with Hyphal Death

In this paper, We studied the case of growth of kinds of fungi when blend two kinds of hyphal anastomosis and Dichotomous branched with a hyphal death. These species consume all energy. We use mathematical models as partial differential equations (PDEs) which illustrate phenomena biological for each kind. we need Some time, that’s true for the growth of fungi. To get an approximate solution for this system, we will rely on the numerical solution. For this, we need Some of the steps in this solution are stationary, phase, and traveling states Solution And to determine the initial condition. we will use the code and Thus we recognize the behavior of the kinds.


1.
Introduction We constructed new models for the development of fungal mycelia. At this scale, partial differential equations representing the interaction of biomass with the underlying substrate are the appropriate choice. These models are of a complex mathematical structure, comprising both parabolic and hyperbolic parts. Thus, their analytic and numerical properties are non-trivial, and for this a group of any number of types can be expressed during the growth stages of a specific type of fungi. To facilitate discussion of these kinds the abbreviated symbols for each type are used, as in table (1), which describe some biological types where each type analyzed mathematically and given an explanation and a description of the parameters. In this paper, we will mix certain types of fungi [1,2,3,4].

Mathematical Model
We will speak about a new kind of fungal branching with fungal death is Tip-hypha anastomosis with dichotomous branching (HY). The table below shows these kinds [6,7]:  2 We are able to describe hyphal growth by using the following system: Where = indicated to the flux and = is hyphal death, and by compensation about ( ( , ) = 1 − 2 ) in above system, we obtain :

Non-dimensionlision and Stability of uniform solution
In this part, we illustrate how can lay these parameters to get dimensionless this parameter is represented "rate of hyphal tips per unit hyphal per unit length hypha per unit time". is hyphal death where the value of = 1 and α (1 − ) indicates to "the number of the tips produced per unit time". Now, we find steady states from the system (3) when take the following: The solution of the system above, we obtain the values of ( , )-plane, the steady state are: (1, 1), (0, 0) .Now, we can take Jacobain for the system(4) We will get two eigenvalues of ( : = 1, 2) We notice through solution the system in the point (0, 0) is unstable node while in the point (1, 1) is the saddle point when is non-negative. The Fig(1) illustrate that by Using "MATLAB pplane7".  Figure 1. The (p, n)plane, we note that trajectories connects the saddle point (1, 1)and the unstable node (0, 0) for α = 2.

Traveling Wave Solution and the steady states
In this part, we will speak the travelling wave solution, let = − , and we impose:

5.
The numerical solution we will be using" pdepe code in MATLAB "to solve the system(3). This showing the numerical solution of the branch ( ) and tips (n), obtain through the initial condition ( 1 → 0). The figures below explains this.

6.
The conclusions we concluded from above results that the traveling wave c increase whenever the values of  increase for time t. Since "the value of α = 1 ," and we notes that α directly proportional with 1 "the number of the tips produced per unit time" and inversely proportional "the rate constant for the hyphal autolysis ". That's mean from a biological point of view the growth increases whenever α increases. . 7.