Periodicity computation of generalized mathematical biology problems involving delay differential equations.

In this paper, we consider a low initial population model. Our aim is to study the periodicity computation of this model by using neutral differential equations, which are recognized in various studies including biology. We generalize the neutral Rayleigh equation for the third-order by exploiting the model of fractional calculus, in particular the Riemann-Liouville differential operator. We establish the existence and uniqueness of a periodic computational outcome. The technique depends on the continuation theorem of the coincidence degree theory. Besides, an example is presented to demonstrate the finding.

Introduction

Biocomputing is proposed as the procedure of constructing models that use biological materials. The class of neutral differential delay equations is the most popular model in Biocomputing. It was introduced by the famous British mathematical biologist, Lord Rayleigh, as follows:Eq. (1) is extended into a third order by various authors. Abou-El-Ela et al. (2012) discussed a criterion for the existence of periodicity to third order neutral delay differential equation with one deviating argument as below:

Using the idea of the fractional calculus (see Podlubny, 1999), Eq. (1) is developed (see Ibrahim et al., 2016a, Ibrahim et al., 2016b, Ibrahim et al., 2016c). Recently, Rakkiyappan et al. (2016) presented the periodicity by applying fractional neural network model.

The objective of this work is to give new appropriate conditions for guaranteeing the existence and uniqueness of a periodic solution of fractional differential equation of order 3μ (0 < μ < 1) with two deviating arguments, taking the formwhere D3μ is the Riemann–Liouville fractional differential operator of order 3μ, and are continuous functions ε1, ε2 and p are periodic, ϑ1 and ϑ2 are periodic in their first argument and T > 0.

Material and methods

For convenience, we let

Let the following setsandare be two Banach spaces with the norms

Outline a linear operator by settingand for ,

We as well term a nonlinear operator by setting

Therefore, we have seen that , ; is a subset of , which implies .

So the operator L is a Fredholm operator with index zero. Now we define a nonlinear operator as follows:where the Riemann–Liouville fractional differential operator is defined as follows:

We need the following outcome:

Method 2.1 (Continuation method) Assume that X and Y be two Banach spaces. Supposing that is a Fredholm operator with index zero and is L-compact on , where is an open bounded subset in X. Furthermore, let the next conditions are satisfied:

The Brower degree

Then has at least one solution on .

Moreover, we need the following assumptions in the sequel:

Suppose that there exist non-negative constants A1; A2; B1; B2; C1 and C2 such as

For all ,

For all and

For all .

Assume that the subsequent conditions are satisfied:

(H1) One of the next conditions holds

(H) There exists d > 0 like one of the following conditions holds

If u(t) is a periodic solution of (6), then

Assume that (i) and (ii) hold such that

If u(t) is a periodic solution of (3), then

Assume that (i)–(iii) hold. Also let the next condition holds

Results

We impose the periodicity computation of the generalized neutral equation (3) in the following result:

Result 3.1: Assume that (i) - (iv) hold. Then (3) has a unique periodic solution.

Demonstration: Condition (iv) implies that (3) has at most one periodic solution. Therefore, it is enough to prove that (3) has at least one periodic solution. Let the set of periodic solutions of (6) be bounded (by the boundedness of ).

Let u(t) be T-periodic solution of (6). Multiplying (4) by and then by integrating it over [0; T], we get

Using the fractional Taylor series (Tarasov, 2016), according to the condition (i), inequality (7) and the Cauchy–Schwarz inequality, we obtain

Thus, there exists such as with the following inequalities:

Let , now we have as a non-empty open bounded subset of X. Thus, condition (a) in Method 2.1 holds. According to (H2)(1) and (H2)(2), we aim to study two cases:

Case (i): Let (H2)(1) hold. Since

For any then u is a constant with . Then

The condition (b) of Method 2.1 is fulfilled. Moreover, we define a continuous function H(u,h)by setting

According to (10) we have uH(u,h) < 0 for all and h ∊ [0,1] therefore, H(u,h) is a homotopic transformation. From the homotopy invariance theorem we get

Therefore, condition (c) is satisfied.

Case (ii): Let (H2)(2) hold. Since

For any

We have

This means that the condition (b) of Method 2.1 is gratified. Define

According to (11) we have and h ∊ [0,1]. Hence, H(u,h) is a homotopic transformation. Using the homotopy invariance theorem we find

Therefore, the condition (c) of Method 2.1 is achieved. Moreover, we conclude that (3) has at least one periodic solution and the solution is unique. This completes our result.

Discussion

To discuss our results, we apply the main Result 3.1, to obtain a periodic solution. Let us consider the -periodic solution of the fractional third-order delay differential equation with two deviating argumentswhere

By (12) we have

Observing that

Case I. (d ⩽ 1) We let , (d is an arbitrary small positive constant). Then we attain

When μ = 1/3, we have κ = 0.519. In addition, When μ = 1/4, we have κ = 0.45, while κ > 1, when μ ⩾ 1/2. In this case, we conclude that 0 < μ ⩽ 1/3. Similarly for the following quantity, when μ = 1/3:

It is understandable that all the assumptions (ii)–(iv) are satisfied. Thus, by our Result 3.1, Equation (12) has a unique -periodic solution.

Case II. (d > 1) When d = 5, μ = 1/3, we have (in this case there is a periodic solution), while when d = 35, μ > 1/3, we have (there is no periodic solution).

Conclusion

The advantage of a periodic solution appears when the initial population is low and this situation may appear in the class of fractional differential equations. We generalized a class of neutral third order differential equations by applying the idea of fractional calculus. The fractional differential operator is taken in the sense of Riemann–Liouville calculus. We imposed the periodicity computation of solutions. The method depends on the continuation theory. We showed by example of a population model, that the periodic solution of the fractional differential equation is approximated to the ordinary equation when the fractional power satisfies the inequality 0 < μ ⩽ 1/3. For future work, one can consider n- deviating arguments depending on the actions of the population growth.

Competing interest

The authors declare that there are no competing interests.