Qualitative Analysis of a Retarded Mathematical Framework with Applications to Living Systems

Abstract

This paper deals with the derivation and the mathematical analysis of an autonomous and nonlinear ordinary differential-based framework. Specifically, the mathematical framework consists of a system of two ordinary differential equations: a logistic equation with a time lag and an equation for the carrying capacity that is assumed here to be time dependent. The qualitative analysis refers to the stability analysis of the coexistence equilibrium and to the derivation of sufficient conditions for the existence of Hopf bifurcations. The results are of great interest in living systems, including biological and economic systems.

1. Introduction

In the last three decades, the mathematical modeling of living systems has gained much attention and different mathematical methods have been developed in an attempt to obtain a mathematical theory. Living systems, differently from physical systems, require more attention and a different mathematical treatment depending on the system under investigation.

Different mathematical and also computational frameworks have been proposed whose difficulty is strictly related to the phenomena of the system that has to be modeled. Indeed, if we are only interested in the time evolution of the density of the elements in the system (no spatial dynamics), the framework of the ordinary differential equations can be employed; see the review paper [1]. It is worth stressing that this framework requires the definition of a differential equation for each element of the system; therefore, if the number of elements of the system is very large, the applicability of this framework may be unfeasible. When the latter occurs, a continuum mechanics approach can be performed; this approach consists in deriving mass, momentum, and energy conservation equations by phenomenological models; see the book [2, 3]. The description is obtained by means of suitable partial differential equations. An intermediate approach is that proposed by the kinetic theory. This approach allows the derivation of integrodifferential equations by considering the role of the interactions that can be conservative, nonconservative, and mutative; see the recent review paper [4].

The time evolution of the elements of the system modeled with the previously mentioned mathematical frameworks has in common the characteristic that the description at a time 𝑡 depends on the description at the same time 𝑡. This is an approximation of the reality because the phenomena that occur at a time 𝑡 are strictly related to the behavior of the system at a previous time 𝑡−𝜏, where 𝜏 is a time lag. Therefore, recently the time delay has been inserted into mathematical models for the biological and economical systems, especially in the ordinary differential equations-based models; see, among others, the review paper [5] and papers [6–14].

This paper is devoted to the definition of a mathematical framework that can be proposed for the modeling of living systems, especially of biological and economical nature. The framework proposed in the present paper is based on ordinary differential equations. Specifically, the mathematical framework consists of a system of two nonlinear ordinary differential equations: a logistic equation with a time lag (see papers [15, 16]) and an equation for the carrying capacity that is assumed to be time dependent. The qualitative analysis refers to the stability analysis of the coexistence equilibrium solution and to the derivation of sufficient conditions for the existence of Hopf bifurcations.

The rest of the paper is organized as follows. After this introduction, Section 2 deals with the definition of the assumptions of the system to be modeled and the derivation of the relative model. Section 3 focuses on the qualitative analysis of the framework and specifically is concerned with the coexistence equilibrium solution and the related stability analysis including sufficient conditions for the existence of a Hopf bifurcation. The quality of the Hopf bifurcation, the stability of the bifurcating periodic trajectory, and the relative period are investigated in Section 4. Finally, we conclude the paper and present applications and research perspectives in Section 5.

2. The Underlying Mathematical Framework

This section is concerned with the derivation of the mathematical framework that can be proposed for the modeling of biological and economic systems.

Specifically, we consider two cooperating populations whose density is denoted by 𝜅=𝜅(𝑡) and 𝑛=𝑛(𝑡), respectively. Furthermore, we assume that the population 𝑛 is defined by a logistic term with time delay 𝜏 such that the carrying capacity is time dependent and coincides with 𝜅. Let 𝑟 be the cooperation rate of population 𝑛, 𝑐 the cooperation rate of population 𝜅, and 𝜆 the decrease rate of the population 𝜅. Bearing all the above assumptions in mind, the mathematical models thus reads as follows 𝜅̇(𝑡)(𝑡)𝜅(𝑡)(𝑡),𝑛(̇𝑡)(𝑡)𝑛(𝑡𝑛()=𝑐𝑛−𝜆𝜅=𝑟𝜅)1−𝑡−𝜏𝜅(𝑡). (1)

The mathematical framework (1) thus consists of a system of autonomous nonlinear ordinary differential equations with a time delay and is characterized by three nonnegative parameters that have specific meanings and can be tuned with empirical data.

It is worth stressing that the mathematical framework (1) summarizes different mathematical models presented in the literature. Indeed, if 𝜏=𝜆=0, we obtain the models analyzed in papers [17, 18] where technological innovation is the driving force behind human population growth; see also papers [19, 20] where the authors assume a positive feedback between technology and population; for 𝜏=0 and 𝜆=𝑟, we obtain the model analyzed in paper [21].

3. Equilibrium Points and Stability Analysis

This section is devoted to analytical investigations on the existence and stability analysis of the equilibria of the mathematical model (1). The following proposition states the number of nontrivial equilibrium points.

Proposition 1. The mathematical model (1) admits the unique nontrivial equilibrium point (𝜅_∗,𝑛_∗)=(𝜆/𝑐,𝜆/𝑐).

Proof. The proof of this proposition follows by observing that equilibrium points of system (1) correspond to solutions of the algebraic system 𝜅̇=𝑛̇=0. Since the mathematical model (1) is composed of autonomous differential equations, then the equilibrium points of the system (1) coincide with those of the undelayed counterpart, namely, when 𝜏=0.

Straightforward calculations show that the characteristic equation of the linearized system at (𝜅_∗,𝑛_∗) of the mathematical model (1) model reads as follows: 𝜉²+𝑟𝑛_∗𝜉𝑒^−𝜉𝜏−𝑐𝑟𝑛²_∗=0, (2) where, for notational convenience, we have 𝑛_∗=𝜆/𝑐.

The stability analysis of the nontrivial equilibrium (𝜅_∗,𝑛_∗) will be performed by examining first the undelayed system. Thus, if there is no time delay, (2) reads as follows: 𝜉²+𝑟𝑛_∗𝜉−𝑐𝑟𝑛²_∗=0, (3) and the associated eigenvalues are 𝜉_1,2=−𝑟𝑛_∗±𝑛_∗√𝑟²+4𝑐𝑟2. (4) Looking to formula (4), we have two real eigenvalues with opposite signs. Then, the equilibrium point (𝜅_∗,𝑛_∗) is a saddle point (system instable).

Suppose now that 𝜏>0. The instability of the equilibrium point may change when the characteristic equation of the system under consideration has zero or a pair of pure imaginary eigenvalues. The former case cannot happen since it occurs if 𝜉=0 in (2) and it is immediately seen that this leads to the contradiction −𝑐𝑟𝑛²_∗=0. Now, we study when (2) has pure imaginary roots 𝜉=±𝑖𝜔, where 𝜔 is a positive real number.

Indeed, if 𝜉=𝑖𝜔, 𝜔>0, is a root of (2), then the following identity must be true: −𝜔²+𝑟𝑛_∗𝑖𝜔𝑒^{−𝑖𝜔𝜏}−𝑐𝑟𝑛²_∗=0, (5) and distinguishing between the real and imaginary parts, we obtain the following two equations: 𝑟𝑛_∗()()𝜔𝜔cos𝜔𝜏=0⟹cos𝜔𝜏=0,₂+𝑐𝑟𝑛²_∗=𝑟𝑛_∗𝜔sin(𝜔𝜏). (6) The solution of (6) is 𝜔𝜏=𝜋/2 (𝜔𝜏=(3𝜋/2), in particular the other solution has been excluded because it implies the absurd 0<𝜔²+𝑐𝑟𝑛²_∗=𝑟𝑛_∗𝜔(−1)<0) and 𝜔²−𝑟𝑛_∗𝜔+𝑐𝑟𝑛²_∗=0, (7) whose solutions read as 𝜔=𝑟𝑛_∗±𝑟𝑛_∗²−4𝑐𝑟𝑛²_∗2=𝑛_∗𝑟±√𝑟²−4𝑐𝑟2. (8) The following lemma holds.

Lemma 2. Let 𝑟 be the population growth rate of population 𝑛 and 𝑐 the cooperation rate of population 𝜅.(1)If 𝑟−4𝑐<0 holds true, then (7) does not admit positive roots. Then, there is no stability switch for system (1). Moreover, since (𝜅_∗,𝑛_∗) is unstable when 𝜏=0, it remains unstable for all 𝜏>0.(2)If 𝑟−4𝑐≥0 holds true, then (7) has a unique positive root 𝜔₀, if 𝑟−4𝑐=0, and admits two different positive roots 𝜔_±, respectively, if 𝑟−4𝑐>0. As 𝜏 increases, stability switches may occur.

It is worth noting that it may be seen easily that the purely imaginary roots 𝜔₀ and 𝜔_± are simple. Moreover, from (6), we also obtain a sequence of the critical values of 𝜏 defined as follows: 𝜏⁰_𝑗=𝜋2𝜔₀+2𝜋𝑗𝜔₀,𝜏^±_𝑗=𝜋2𝜔_±+2𝜋𝑗𝜔_±𝑗=0,1,2,…. (9)

Lemma 3. Let 𝑟 be the population growth rate of population 𝑛 and 𝑐 the cooperation rate of population 𝜅.(1)If 𝑟²−4𝑐𝑟=0, then (2) with 𝜏=𝜏⁰_𝑗 has a pair of simple pure imaginary roots ±𝑖𝜔₀.(2)If 𝑟²−4𝑐𝑟>0, then (2) with 𝜏=𝜏^±_𝑗 has two pairs of simple pure imaginary roots ±𝑖𝜔_±.

3.1. Existence of Hopf Bifurcations

In this subsection, sufficient conditions for the existence of a Hopf bifurcation are stated. The preliminary result is contained in the following lemma.

Lemma 4. Let 𝜉(𝜏)=𝜈(𝜏)+𝑖𝜔(𝜏) denote the roots of (2) near 𝜏=𝜏_∗ satisfying 𝜈(𝜏_∗)=0, 𝜔(𝜏_∗)=𝜔_∗, where 𝜏_∗∈{𝜏⁰_𝑗,𝜏^±_𝑗}(𝑗=0,1,2,…) and 𝜔(𝜏_∗)∈{𝜔₀,𝜔_±}, with 𝜏⁰_𝑗,𝜏^±_𝑗 and 𝜔₀,𝜔_± being as defined in (8) and (9), respectively. (1)Let 𝑟−4𝑐=0, so that 𝜔_∗=𝜔₀. One has 𝑑[Re𝜉(𝜏_∗)]/𝑑𝜏=0. Hence, the transversality condition 𝑑[Re𝜉(𝜏_∗)]/𝑑𝜏≠0 does not hold. (2)Let 𝑟−4𝑐>0. (a)If 𝜔_∗=𝜔₊, then the transversality condition reads 𝑑Re𝜉𝜏_∗𝑑𝜏>0. (10) (b)If 𝜔_∗=𝜔₋, then the transversality condition reads 𝑑Re𝜉𝜏_∗𝑑𝜏<0. (11)

Proof. Differentiating (2) with respect to 𝜏 and using (2), we get 𝑑𝜉=𝑑𝜏𝑐𝑟𝑛²_∗−𝜉²𝜉²2𝜉²+𝑐𝑟𝑛²_∗−𝜉²(1−𝜉𝜏). (12) Substituting 𝜉=𝑖𝜔_∗ into (12), flipping it over, and taking its real part, one has Re𝑑𝜉𝑑𝜏⁻¹_𝜏=𝜏∗=2𝑐𝑟𝑛²_∗+𝜔²_∗−1𝜔²_∗. (13) Using (7), then (13) reads 𝜉Re𝑑𝜏⁻¹_𝜏=𝜏∗=𝜔²_∗−𝑐𝑟𝑛²_∗𝑐𝑟𝑛²_∗+𝜔²_∗𝜔²_∗=2𝜔_∗−𝑟𝑛_∗𝑐𝑟𝑛²_∗+𝜔²_∗𝜔_∗. (14) Therefore, 𝑑signRe𝜉(𝜏)𝑑𝜏_𝜏=𝜏∗=signRe𝑑𝜉𝑑𝜏⁻¹_𝜏=𝜏∗=sign2𝜔_∗−𝑟𝑛_∗. (15) Then, from (7), we have the following results.(i)Let 𝑟−4𝑐=0; then 𝜔_∗=𝜔₀, so that sign{2𝜔_∗−𝑟𝑛_∗}=0.(ii)Let 𝑟−4𝑐>0. If 𝜔_∗=𝜔₊, then sign{2𝜔_∗−𝑟𝑛_∗}=sign{𝑛_∗√𝑟²−4𝑐𝑟}=+1. If 𝜔_∗=𝜔₋, then sign{2𝜔_∗−𝑟𝑛_∗}=sign{−𝑛_∗√𝑟²−4𝑐𝑟}=−1.This completes the proof.

Bearing the above analysis in mind, we have that the root 𝜉(𝜏) of (2) crosses the imaginary axis from left to right at 𝜏=𝜏⁺_𝑗 (because 𝑑[Re𝜉(𝜏⁺_𝑗)]/𝑑𝜏>0) and from right to left at 𝜏=𝜏⁻_𝑗 (because 𝑑[Re𝜉(𝜏⁻_𝑗)]/𝑑𝜏<0), as 𝜏 increases.

Summarizing the above remarks and combining the lemmas, we have the following results on the distribution of roots of (2).

Theorem 5. For the system (1), the following statements hold true.(1)If 𝑟−4𝑐<0, the equilibrium (𝜅_∗,𝑛_∗)=(𝜆/𝑐,𝜆/𝑐) is unstable for all 𝜏≥0.(2)If 𝑟−4𝑐>0, we have two cases:(a)the equilibrium (𝜅_∗,𝑛_∗) is unstable for all 𝜏≥0 and undergoes a Hopf bifurcation at (𝜅_∗,𝑛_∗) when 𝜏=𝜏⁺_𝑗(𝑗=0,1,2,…);(b)the equilibrium (𝜅_∗,𝑛_∗) is unstable for 𝜏∈[0,𝜏⁻₀) and stable for 𝜏>𝜏⁻₀. Moreover, it undergoes a Hopf bifurcation at (𝜅_∗,𝑛_∗) when 𝜏=𝜏⁻_𝑗(𝑗=0,1,2,…).

Remark 6. In the previous theorem, the dynamics proprieties of the mathematical model are missing when 𝑟−4𝑐=0. Indeed, in this case, the transversality condition disappears.

4. Qualitative Analysis of the Hopf Bifurcation

Following the normal form method and the center manifold theory [22], this section is concerned with the derivation of explicit formulas for determining the properties of the Hopf bifurcation (it exists when 𝑟−4𝑐>0, see Theorem 5) at critical value 𝜏_∗=𝜏⁻_𝑗 or 𝜏⁺_𝑗, where 𝜆(𝜏_∗)=𝑖𝜔_∗.

In what follows, we denote by 𝐶=𝐶([−𝜏_∗,0],ℝ²) the space of the continuous vector function 𝑣∶−𝜏_∗,0⟶ℝ², (16) and by 𝐶=𝐶([0,𝜏_∗],ℝ²) the space of the continuous vector function 𝑣∶0,𝜏_∗⟶ℝ². (17) Setting 𝜇=𝜏−𝜏_∗∈ℝ and applying to the mathematical model (1) the change of variables 𝑢₁=𝜅−𝜅_∗,𝑢₂=𝑛−𝑛_∗, (18) the model is equivalent to the functional differential equation system: 𝑢̇(𝑡)=𝐿_𝜇𝑢_𝑡+𝑓𝜇,𝑢_𝑡, (19) where(i)𝑢=(𝑢₁,𝑢₂),(ii)𝑢_𝑡(𝜃)=𝑢(𝑡+𝜃) for 𝜃∈[−𝜏_∗,0],(iii)𝐿_𝜇∶𝐶→ℝ² reads as 𝐿_𝜇𝜑=𝐵₁𝜑(0)+𝐵₂𝜑−𝜏_∗, (20)where 𝐵₁=0𝑐𝑛_∗𝑟𝑛_∗0,𝐵₂=000−𝑟𝑛_∗, (21)(iv)𝑓∶ℝ×𝐶→ℝ² reads as 𝑓=𝑓𝜇,𝜑¹𝑓₂, (22)where 𝑓₁=𝑐𝜑₁(0)𝜑₂(0),𝑓₂=𝑟𝜑₁(0)𝜑₂(0)−𝑟𝜑₂(0)𝜑₂−𝜏_∗. (23)

Lemma 7. Let 𝐿_𝜇(𝜑) be the vector function defined by (20). Then, there exists a 2×2 matrix-valued function of bounded variation 𝜂(𝜃,𝜇), for 𝜃∈[−𝜏_∗,0], such that 𝐿_𝜇𝜑=⁰_−𝜏∗𝜑(𝜃)𝑑𝜂𝜃,𝜇. (24)

Proof. The proof is obtained by applying the Riesz representation theorem. Indeed, we may take 𝜂𝜃,𝜇=𝐵₁𝛿(𝜃)+𝐵₂𝛿𝜃+𝜏_∗, (25) where 𝛿(𝜃) is the Dirac delta function.

Definition 8. Let 𝜂(𝜃,𝜇) be the 2×2 matrix-valued function 𝜂(𝜃,𝜇) of bounded variation of Lemma 7. For 𝜑∈𝐶, one defines the following operators: 𝐴𝜇𝜑(𝜃)∶=𝑑𝜑𝑑𝜃,𝜃∈−𝜏_∗,0_0−𝜏∗,𝜑(𝑠)𝑅𝜇𝜑𝑑𝜂𝑠,𝜇,𝜃=0,∶=0,𝜃∈−𝜏_∗,𝑓,0𝜇,𝜑,𝜃=0. (26)(27)

Remark 9. A straightforward analysis shows that the system (19) is equivalent to 𝑢̇(𝑡)=𝐴𝜇𝑢_𝑡+𝑅𝜇𝑢_𝑡, (28) where 𝐴(𝜇)(𝜑) and 𝑅(𝜇)(𝜑) are the operators (26) and (27), respectively, and 𝑢_𝑡=𝑢(𝑡+𝜃), for 𝜃∈[−𝜏_∗,0].

Definition 10. Let 𝜂(𝜃,𝜇) be the 2×2 matrix-valued function 𝜂(𝜃,𝜇) of bounded variation of Lemma 7. For 𝜓∈𝐶=𝐶([0,𝜏_∗],ℝ²), one defines the following operator: 𝐴^∗𝜇𝜓(𝑠)=−𝑑𝜓(𝑠)𝑑𝑠,𝑠∈0,𝜏_∗,_0−𝜏∗𝑑𝜂𝜁,𝜇𝜓(−𝜁),𝑠=0, (29) and one defines the following bilinear form: 𝜓(𝑠),𝜑(𝜃)=𝜓(0)𝜑(0)−⁰_{𝜃=−𝜏∗}^𝜃_𝜎=0𝜓(𝜎−𝜃)𝑑𝜂(𝜃)𝜑(𝜎)𝑑𝜎, (30) where 𝜂(𝜃)=𝜂(𝜃,0).

Remark 11. It is easy to show that 𝐴(0) and 𝐴^∗(0) are adjoint operators. We know that ±𝑖𝜔_∗ is an eigenvalue of 𝐴(0); then ±𝑖𝜔_∗ is also an eigenvalue of 𝐴^∗(0).

Let 𝑞(𝜃) and 𝑞^∗(𝑠) be the eigenvectors for 𝐴(0) and 𝐴^∗(0) corresponding to 𝑖𝜔_∗ and −𝑖𝜔_∗. Assume that 𝑞(𝜃)=𝑞(0)𝑒^{𝑖𝜔∗𝜃}=(1,𝛼₁)𝑒^{𝑖𝜔∗𝜃}, with 𝛼₁∈ℂ, is the eigenvector of 𝐴(0) corresponding to 𝑖𝜔_∗. From 𝐴(0)𝑞(𝜃)=𝑖𝜔_∗𝑞(𝜃), we have (𝐵₁−𝑖𝜔_∗𝐼+𝐵₂𝑒^{−𝑖𝜔∗𝜃})𝑞(0)=0, so that we can evaluate 𝛼₁. Similarly, we assume 𝑞^∗(𝑟)=𝑞^∗(0)𝑒^{𝑖𝜔∗𝑟}=𝐷(𝛼₂,1)𝑒^{𝑖𝜔∗𝑟}, where 𝛼₂ and 𝐷 are complex values, and get (𝐵^𝑇₁+𝑖𝜔_∗𝐼+𝐵^𝑇₂𝑒^{−𝑖𝜔∗𝜃})𝑞^∗(0)=0. In this way, we can find 𝛼₂ with the value of 𝐷 chosen so that ⟨𝑞^∗(𝑠),𝑞(𝜃)⟩=1.

We will use now the theory by Hassard et al. [22] to compute the coordinates describing the center manifold 𝒞 at 𝜇=0. Let 𝑢_𝑡 be a solution of (28) with 𝜇=0 and 𝑧(𝑡)∶=𝑞^∗,𝑢_𝑡,𝑊(𝑡,𝜃)∶=𝑢_𝑡(𝜃)−2Re𝑧(𝑡)𝑞(𝜃). (31) On the center manifold 𝒞, one has 𝑊(𝑡,𝜃)=𝑊(𝑧(𝑡),𝑧(𝑡),𝜃), (32) where 𝑊(𝑡,𝜃)=𝑊(𝑧(𝑡),𝑧(𝑡),𝜃)=𝑊₂₀(𝜃)𝑧²2+𝑊₁₁(𝜃)𝑧𝑧+𝑊₀₂(𝜃)𝑧²2+⋯, (33) and 𝑧 and 𝑧 are local coordinates for the center manifold 𝒞 in the direction of 𝑞 and 𝑞. For the solution 𝑢_𝑡∈𝒞 of (28), since 𝜇=0, we get 𝑧(̇𝑡)=𝑖𝜔𝜏_∗𝑧+𝑞^∗(0)𝑓₀(𝑧,𝑧), (34) where 𝑓₀(𝑧,𝑧)=𝑓0,𝑊(𝑧,𝑧,0)+2Re𝑧(𝑡)𝑞(0), (35) and 𝑊(𝑧,𝑧,0) is given by (32). We rewrite this equation 𝑧(̇𝑡)=𝑖𝜔_∗𝑧+𝑔(𝑧,𝑧), where 𝑔(𝑧,𝑧)=𝑞^∗(0)𝑓₀(𝑧,𝑧)=𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2+⋯. (36) Since 𝑢_𝑡(𝜃)=𝑊(𝑡,𝜃)+𝑧𝑞(𝜃)+𝑧𝑞(𝜃), substituting into 𝑓₀, we find 𝑓₀=𝑓_𝑧2𝑧²2+𝑓_𝑧𝑧𝑧𝑧+𝑓_𝑧²𝑧²2+𝑓_𝑧2𝑧𝑧²𝑧2+⋯. (37) Comparing the coefficients between (36) and (37), we obtain 𝑔₂₀=𝑞^∗(0)𝑓_𝑧2,𝑔₀₂=𝑞^∗(0)𝑓_𝑧²,𝑔₁₁=𝑞^∗(0)𝑓_𝑧𝑧,𝑔₂₁=𝑞^∗(0)𝑓_𝑧2𝑧. (38) We remark that 𝑔₂₁, will depend on 𝑊₂₀(𝜃) and 𝑊₁₁(𝜃). Hence, in order to determine 𝑔₂₁, we need to compute them. By (28) and (36), we have 𝑊̇=𝑢̇_𝑡−𝑧𝑞̇−𝑧̇𝑞=𝐴(0)𝑊−2Re𝑞^∗(0)𝑓₀𝑞(𝜃),𝜃∈−𝜏_∗,𝐴(0),0𝑊−2Re𝑞^∗(0)𝑓₀𝑞(0)+𝑓₀(0)(,𝜃=0,∶=𝐴𝑊+𝐻𝑧,𝑧,𝜃), (39) where 𝐻(𝑧,𝑧,𝜃)=𝐻₂₀(𝜃)𝑧²2+𝐻₁₁(𝜃)𝑧𝑧+𝐻₀₂(𝜃)𝑧²2+⋯. (40) Expanding the above series and comparing the corresponding coefficients of 𝑧², 𝑧𝑧, and 𝑧², we obtain 𝐴(0)−2𝑖𝜔_∗𝑊₂₀(𝜃)=−𝐻₂₀(𝜃),𝐴(0)𝑊₁₁(𝜃)=−𝐻₁₁(𝜃). (41) From (39), for 𝜃∈[−𝜏_∗,0), we can get 𝐻(𝑧,𝑧,𝜃)=2Re𝑞^∗(0)𝑓₀𝑞(𝜃)=−𝑞(𝜃)−𝑔𝑞(𝜃). (42) Comparing the coefficients with (40), we obtain 𝐻₂₀(𝜃)=−𝑔₂₀𝑞(𝜃)−𝑔₀₂𝑞(𝜃),𝐻₁₁(𝜃)=−𝑔₁₁𝑞(𝜃)−𝑔₁₁𝑞(𝜃). (43) Combining (41) and (43), we derive 𝑊̇₂₀(𝜃)=2𝑖𝜔_∗𝑊₂₀(𝜃)−𝐻₂₀(𝜃),𝑊̇₁₁(𝜃)=−𝐻₁₁(𝜃). (44)(45) Therefore, 𝑊₂₀(𝜃)𝑔=−²⁰𝑖𝜔_∗𝑞(0)𝑒^{𝑖𝜔∗𝜃}−𝑔₀₂3𝑖𝜔_∗𝑞(0)𝑒^{−𝑖𝜔∗𝜃}+𝐸₁𝑒^{2𝑖𝜔∗𝜃},𝑊₁₁(𝜃)=𝑔₁₁𝑖𝜔_∗𝑞(0)𝑒^{𝑖𝜔∗𝜃}−𝑔₁₁𝑖𝜔_∗𝑞(0)𝑒^{−𝑖𝜔∗𝜃}+𝐸₂, (46)(47) where (𝐸₁,𝐸₂) is a constant vector to be determined. From (31), we find 𝑑𝜂(⁰_−𝜏∗𝜃)𝑊₂₀(𝜃)=2𝑖𝜔_∗𝑊₂₀(𝜃)−𝐻₂₀(𝜃),_0−𝜏∗𝑑𝜂(𝜃)𝑊₁₁(𝜃)=−𝐻₁₁(𝜃). (48)(49) From (31) again, for 𝜃=0, we see that 𝐻₂₀(0)=−𝑔₂₀𝑞(0)−𝑔₀₂𝑞(0)+𝑓_𝑧2,𝐻₁₁(0)=−𝑔₁₁𝑞(0)−𝑔₁₁𝑞(0)+𝑓_𝑧𝑧. (50)(51) Substituting (46) and (50) into (48) and observing that 𝑖𝜔_∗𝐼−⁰_−𝜏∗𝑒^{𝑖𝜔∗𝜃}𝑑𝜂(𝜃)𝑞(0)=0,−𝑖𝜔_∗𝐼−⁰_−𝜏∗𝑒^{−𝑖𝜔∗𝜃}𝑑𝜂(𝜃)𝑞(0)=0, (52)(53) we have 2𝑖𝜔_∗−⁰_−𝜏∗𝑒^{2𝑖𝜔∗𝜃}𝑑𝜂(𝜃)𝐸₁=𝑓_𝑧2, (54) and then we get 𝐸₁. Similarly for 𝐸₂, we have 𝑑𝜂(⁰_−𝜏∗𝜃)𝐸₂=𝑓_𝑧𝑧. (55) Thus, we can calculate all the four coefficients 𝑔_𝑖𝑗 and so can determine the following quantities, which are required for the analysis of Hopf bifurcation: 𝐶₁(0)=𝑖2𝜔_∗𝑔₁₁𝑔₂₀−2𝑔₁₁²−𝑔₀₂²3+𝑔₂₁2,𝜇₂𝐶=−Re₁(0)Re𝜆^′𝜏_∗,𝛽₂=2Re𝐶₁(0,𝑇₂)𝐶=−Im₁(0)+𝜇₂Im𝜆^′𝜏_∗𝜔_∗. (56) The periodic solutions and their stabilities can be analyzed with the aid of the normal form parameters 𝐶₁(0), 𝜇₂, 𝛽₂, and 𝑇₂. In particular, 𝜇₂ allows us to determine the quality of the Hopf bifurcation (supercritical or subcritical); 𝛽₂ determines the stability of the bifurcating periodic solutions; 𝑇₂ determines the period of the bifurcating periodic solutions.

Theorem 12. Let (𝜅_∗,𝑛_∗) be the equilibrium point of the model (1).(1)If 𝜇₂>0, then the Hopf bifurcation of system (1) at the equilibrium point (𝜅_∗,𝑛_∗), when 𝜏=𝜏_∗, is supercritical; if 𝜇₂<0, the Hopf bifurcation is subcritical.(2)The bifurcating periodic solutions are locally asymptotically stable if 𝛽₂<0 and unstable if 𝛽₂>0.(3)The period of the bifurcating periodic solutions increases if 𝑇₂>0 and decreases if 𝑇₂<0.

5. Applications and Research Perspectives

The mathematical framework proposed in the present paper can be used in applications. Indeed, the framework can be applied for the modeling of biological system, such as the competition between cancer cells and immune system cells; see paper [23]. The logistic growth term models the maximum cancer cells population size that can be sustained during proliferation. Moreover applications refer to economic growth theory; see [24–26]. the logistic growth is usually attributed to resources per individual becoming scarcer as population size increases. In the case of human population growth, however, it is natural to assume that technology, social organization, and other aspects of culture have allowed humans to increase the environment's carrying capacity.

The mathematical framework proposed in this paper is certainly worthy of future research concerning both its qualitative analysis and the application to modeling complex systems in applied sciences. Specifically, analytical investigations can be addressed to the existence of limit cycles [27–29] such as the contents of the paper [30]. Moreover, it is a research perspective to analyze the stability and Hopf bifurcation of the model (1) when time delay is inserted also in the 𝜅 function.

Finally, the mathematical framework (1) can be generalized in order to take into account strategies performed by the elements of the system (especially in the context of biological systems) by using methods of thermostated kinetic equations [31, 32] that allow the possibility to reach a stationary state [33]. Moreover, this approach permits us to perform asymptotic limits linking ordinary differential equations with kinetic and continuum mechanics approaches [34].

Acknowledgment

The first author acknowledges the support of the FIRB project RBID08PP3J-Metodi Matematici e Relativi Strumenti per la Modellizzazione e la Simulazione della Formazione di tumori, Competizione con il Sistema Immunitario, e Conseguenti Suggerimenti Terapeutici.