Tritrophic Food Chain Model

MODEL FORMULATION

The tritrophic food chain model of biocontrol of Lesser date moth in Palm tress can be written as a set of three coupled nonlinear ordinary differential equations as follows [26]:

((dX)/(dt)) = rX(1-(X/K))-((bXY)/(a+X)), X(0)>0; ((dY)/(dt)) = -dY+((mXY)/(a+X))-pYZ, Y(0)>0; #2.1 ((dZ)/(dt)) = -μZ+qYZ; Z(0)>0;

Fractional order models are more accurate than integer-order models as fractional order models allow more degrees of freedom. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. The presence of memory term in such models not only takes into account the history of the process involved

where F⋅V⁻¹is the next generation matrex for the model (2.2). It the follows that the spectral radius of matrex F⋅V⁻¹ is ρ(F⋅V⁻¹)=max(λ_{i}), i=1,2. Then R₀=((m K)/(d(a+K))). ■ 3. By (2.2), The third point is a positive quilibrium point E₂=(X₂,Y₂,Z₂)=(((a d)/(m-d)),((r a m d (a+K)(R₀-1))/(Kb(m-d)²)),0) , which is the free predator whose population density Z (Z₂=0). 4. The fourth point is the endemic from all types of infection. Then (X≠0≠Z). Therefore the third equilibrium point is E₃=(X₃,Y₃,Z₃) where

X₃ = (1/2)(R+K-a), Y₃ = (μ/q), Z₃ = (((R+K)(m-d)-a(m+d))/(p(R+K+a))), R = √((K+a)²-((4Kbμ)/(rq))).

Remark 1. 1) The free pest whose population density is denoted by Y, (Y=0 and Z=+ve), does not exist, being a natural enemy of the concerned pest, prey on Y and only on Y, so if Y=0 then it should be that, Z=0 is the free equilibrium point E₁ again. 2) E₂ must be have non negative component, then we have the condition m>d and R₀>1 d for E₂. 3) E₃ must be have non negative component, then we have the condition a r q>b μ, (R+K)(m-d)>a(m+d) for

We have obtained a stability condition for equilibrium points. We have also given a numerical example and verified our results. One should note that although the equilibrium points are the same for both integer order and fractional order models, the solution of the fractional order model tends to the fixed point over a longer period of time. One also needs to mention that when dealing with real life problems, the order of the system can be determined by using the collected data. The transformation of a classical model into a fractional one makes it very sensitive to the order of differentiation α : a small change in α may result in a big change in the final result. From the numerical results Figures follows, it is clear that the approximate solutions depend continuously on the fractional derivative α. We use some documented data for some parameters like r=1, K=2, a=0.9, b=1, d=1, m=3, p=1, μ=1, q=0.5 and (X₀,Y₀,Z₀)=(0.5,1,0.05) in figure 1. And r=1, k=2, a=1, b=1, d=1, m=3, p=1, μ=1, q=1.5 and (X₀,Y₀,Z₀)=(1.4,0.6,0.7) in figure