Mathematical Model Of YFWXHD Branching Type With Hyphal Death

— Mathematical modeling is used to describe the fungus growth process. This model depicts the growth-related behavior of Dichotomous branching, Lateral branching , Tip-tip anastomosis , Tip death due to Overcrowding, Tip-hypha anastomosis with haphal death , we are aware that fungi require money to flourish. Money and effort. Thus, we get a mathematical solution. Although the error ratio, to reduce the time, expense, and work needed to get the right conclusion. In this paper, we will use a system of partial differential equations to solve a mathematical model (PDEs), and for the numerical analysis, we applied several codes, (pplane8, pdepe).


Introduction
There are several papers explaining mathematical models that have been put out by numerous researchers, such as: -In ( 2022) Z. Hussein Khalil and A. Shuaa [7], In this study, researchers investigated the possibility of fungi growing when four different forms of them were combined.These types used all the energy.-In ( 2022) A. Shuaa and A. Saleem Habeeb [8], In this paper, they explain The mathematical model illustrates energy consumption, tip-hypha anastomosis, lateral branching, dichotomous branching, and hyphal death.
-In ( 2022) N. Fawzi Khwedim and A. Hussein Shuaa [9], The mathematical model explains the dysplasia phenomenon.The model that shows the growth of filament tip anastomosis, limb anastomosis, bilateral branching, lateral branching, and limb death owing to crowding with thread death.
In this study, a new model for the growth of fungus is created.Table 1 lists many fungus species, their biological classifications, and versions of these biological phenomena in mathematical form.We combined many types of fungus in this model to explain how the parameters are described.The first person to transform biological events into mathematical form was Leah-Keshet [1].

Haphal death
is the loss rate of hyphal (constant for hyphal death).

Mathematical Model
We will study a new type of branching of fungal growth with hyphal death, the system listed below can be used to describe hyphal growth Where (, ) =  1  +  2  −  1  2 −  3  2 −  2  +  1  Then the system (1) becomes

Non-dimensionlision and Stability
We must reduce the amount of parameters when solving any mathematical system with multiple variables; this process is known as non-dimensionalisation.Ali H. Shuaa Al-Taie (2011) clear up how can put these parameters as dimensionlisionless [2].
Where  = To find steady states for system (3) Then The steady state are (0,0) and Hence we use Jacobian for these equations The steady state are : (0,0) saddle point And ( ) stable spiral ).

Traveling wave solution
We'll discuss the traveling wave solution.Assume that (, ) = (), and (, ) = () where  =  − , () and () are density profile and c rate of propagation of colony edge.() and () non-negative function of z, the function (, ), (, ) are traveling waves and are moves at constant speed c in positive x direction.Where  > 0, and =2, and =1.To look for traveling wave solution of equations in x and t in the form (3).
See [3], therefore the equation above becomes Then the steady state of equation ( 4) is :  ).

Numerical Solution
We use the pdepe code in MATLAP to solve the system (3) numerically because it cannot be solved directly.This paper came to the conclusion that traveling wave solution c and parameter had a relationship where traveling wave increased whenever the values of α increased.View Fig (7).

Conclusion
The wave speed c is clearly increasing when α is an increase function when we plot the connection between c and α, as shown in Fig 7 .Furthermore, we plot the connection between c and β (See Fig. 8), so that the wave speed c decreases as the function β increases.Furthermore, we plot the connection between c and v (See Fig. 9), which clearly shows that the wave speed c is increasing when v is an increase function.In fact, Fig. 10 shows that the wave speed is decreasing when the values of d are increasing.
Since  =  2  ̅   =  3 ̅ 2 ̅ therefore the growth rate is increasing with  2 , while keeping  ̅ are fixed , and the growth rate decreasing with  3 ̅ 2 , while keeping  ̅ fixed, [5,6].In terms of biology, this means that when  grows, the growth also rises.And finally that the growth increases according to  2 increasing , and everytime  3 and ̅ 2 rise, the growth decreases.

Figure l :
Figure l : The (p,n) plane note that The Irajectory Connects the Saddle point (0,0) and the stable spiral ( 2 3 , 2 3

Figure 4 :
Figure 4: The blue line represented tips n.

Figure 5 :
The red line represented branches p.

Figure 6 :
Figure 6: The blue line represented tips n ,with the red line represented branches p.

Figure 9 :
Figure 9: The relation between the wave speed c and values of v.

Table 5 :Figure 10 :
Figure 10: The relation between the wave speed c and the values of d.