# Global Stability of an Economic Model with a Continuous Delay of Kaldor Type Modified

Authors: Aka Fulgence Nindjin; Tetchi Albin N’guessan; Sahoua Hypolithe Okou A Kpetihi; Kessé Tiban Tia
Journal: International Journal of Differential Equations
 (2018)

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## Abstract

This paper studies continuous nonlinear economic dynamics with a continuous delay of a Kaldor type modified in dimension two. The important results are, on the one hand, the boundedness of solutions, the existence of an attractive set, and the permanence of the system and, on the other hand, the local and global stability of equilibrium points.


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## Body

## 1. Model

The complex nonlinear dynamic has been introduced into the analysis of economic phenomena to explain not only the fluctuations observed in the series studies but also the economic crisis in the capitalist system. Thus, economists such as Goodwin (1967) and Kaldor (1955-1956) have employed dynamic models to explain that the cyclic and chaotic growth curves are the economic phenomena endogenous to the capitalist system itself. Within the framework of structural reforms for economic dynamics, NINDJIN et al. in [1] suggest a proven model originating from ecology which could have a certain range in analysis and regulation of financial systems. This model of dimension two describes how the GDP and an economic capital interact in accordance with the model of Kaldor modified in order to increase the resilience of these dynamics against possible disruptions. Mathematical analysis of this model (see [1]) demonstrates that it is bounded and permanent and admits under certain circumstances an attractive set. On the one hand, this permanence manifests itself in the form of stationary growth of capital stock and product (stable interior equilibrium point). On the other hand, it appears in the capital cyclic growth and the product (limit cyclic). So, a capital stock rupture or a long-term production is prevented. It is also shown that the financial system stability (relative to the capital and the product) is overall; i.e., it does not depend on either stock level or production at the initial time. Facing a possible disruption of one of the control parameters of the economic system, we have analyzed, in [2], the model bifurcations. We have shown that the model admits a transcritical bifurcation, a pitchfork bifurcation or a Hopf bifurcation. In the last one, when the model is disrupted, it changes from a stationary balanced growth to a cyclic growth by preventing a GDP or capital crisis. Interactions between the GDP and the economy’s capital could not be clearly and definitively understood or explained regardless of past situations which may affect the present or the future. In this paper, we are going to focus on the way in which investments are evolved and savings are established. Indeed, economies finance their investments through their own savings or those of other economies via financial structures with an interest rate. Regarding the savings, it is known that they are made up of a portion of profits or wages. Thus, when the economy is able to be self-funding by its savings, then the net profit increases πnet=π-rD (i.e., profit, π deprived of the portion set aside to pay interest, r debt, D). So, saving at a time t depends on the net profit; consequently, the GDP from a time t0=t-T¯ where T¯>0. Let us suppose that T¯ is the deadline needed by this saving to reach a certain threshold likely to ensure the self-financing of the investments of the economy.Let us consider the dynamics with no delay of the modified Kaldor type and the following assumption:1.1Y˙=a0Y1-YC0+αa1K+b1Y1-m2KY+C2-αm1YY+C1K,K˙=a2Y+b21-m2KY+C2K-ϱK,Y0>0,K0>0,Y,K∈C10;+∞​;R+,with(a0,C0,C1,C2,m1,m2,α)∈(R+∗)7 and (a1,a2,b1,b2,ϱ)∈(R+)5.Y denotes the product, K denotes the stock of capital, and Y˙ and K˙ indicate, respectively, the growth rate of the product and the stock of capital depending on the following economic parameters:(i)
a0 the trend of rate of increase in GDP for a given (future) period in absence (or in neglect) of the losses,(ii)
C0 the maximum (monetary) value of GDP we can get from this economy for that given period,(iii)
α the currency adjustment factor,(iv)
m1 the maximum (monetary) value of the genuine saving of this economy for the given period,(v)
C1 the maximum (monetary) value of the saving supported by the economy in the given period,(vi)
m2 the maximum value of the investment rates losses for the given period,(vii)
C2 the maximum capital stock for the given period,(viii)
a1 the derivative relative to the capital of the investment rate in the absence (or in the neglect) of the losses (when m2=0) for the given period,(ix)
b1 the investment rate when the capital is null (K=0) and this rate has suffered no loss (m2=0) for the given period,(x)
a2 the share of the GDP converted into stock of capital for the given period,(xi)
b2 the accumulation rate of capital when the product is null (Y=0) and that the investment rate has suffered no loss (m2=0) for the given period.(see [1], page 4).Assumption 1.
The ratio saving-capitalg(Y,K)=m1Y/(Y+C1) at a given time t depends on the GDP produced since t0=t-T¯ with a probability of exponential lag (see, [3], appendix, page 272).So, the ratio saving-capital with delay isg~(t,T¯)=m1x(t)/(x(t)+C1) with x(t)=∫t-T¯tμe-μ(t-ξ)Y(ξ)dξ so that ∫-∞tμe-μ(t-ξ)dξ=1,μ∈R+∗. Then,(1)x˙=-μx-Y-μe-μT¯Yt-T¯.By replacingg(Y,K) by g~(t,T¯), one obtains continuous nonlinear economic dynamics with a continuous delay of a Kaldor type modified in dimension two. Enclosing (1) to the system with a continuous delay, one gets the following system with discrete delay in dimension three:1.2Y˙=a01-YC0Y+αa1K+b11-m2KY+C2Y-αm1xx+C1K,K˙=a2Y+b21-m2KY+C2K-ϱK,x˙=-μx-Y-μe-μT¯Yt-T¯,Y,K,x∈C10;+∞;R+,Y0>0,K0>0,with(a0,a1,a2,b1,b2,c0,c1,c2,m1,m2,ϱ,α,T¯,μ)∈(R+∗)14,Yt(θ)=Y(t+θ), ∀θ∈[-T¯;0], and the feedback function Y0=φ1∈C([-T¯;0];R+) is consequently Y0(0)=Y(0).To facilitate the qualitative study of system1.2 which possesses 14 parameters (a0,a1,a2,b1,b2,c0,c1,c2,m1,m2,ϱ,α,T¯,μ)∈(R+∗)14, let us change the variables by reducing the number of parameters.Let us define the new variables: (2) τ = a 0 t , u τ = Y t c 0 , v τ = m 2 c 0   K t , w t = x c 0 = ∫ t - T t λ e - λ t - ξ u ξ d ξ .Let us define the new parameters of control: (3) β 1 = α a 1 C 0 a 0 m 2 , β 2 = a 2 C 0 a 0 , α 1 = α b 1 a 0 , α 2 = b 2 a 0 , d 1 = C 1 C 0 , d 2 = C 2 C 0 , γ = α m 1 a 0 m 2 , δ = ϱ a 0 , T = a 0 T ¯, and λ=μ/a0. Then, system 1.2 becomes 1.3u˙τ=1-uτuτ+β1vτ+α11-vτuτ+d2uτ-γwτwτ+d1vτ,v˙τ=β2uτ+α21-vτuτ+d2vτ-δvτ,w˙τ=-λwτ-uτ-λe-λTuτ-T,u,v,w∈C10;+∞;R+u0>0,v0>0,w0>0,with(d1,d2,γ,α1,α2,β1,β2,δ,T,λ) ∈ (R+∗)10 and u0 = φ∈C([-T;0];R+) so that u0(0)=u(0).In the long run, we shall adopt the following notations:(4)φ=supθ∈-T;0φθ,k=1-e-λT,andw0=∫-T0λφθeλθdθ.


## 2. Boundedness and Equilibria Points of1.3

### 2.1. Boundedness of the Solutions of Model1.3

Lemma 2.
The interiorint⁡(R+3) and the boundary ∂(R+3) of a positive cone R+3 are invariant for model 1.3.Proof.
Given0≤T0<+∞. Let τ be in [-T;T0[. One knows that (5)uτ=u0exp⁡∫0τ1-ut+β1vt+α11-vtut+d2-γvtwtutwt+d1dt,vτ=v0exp⁡∫0τβ2ut+α21-vtut+d2-δdt,wτ=w0exp⁡∫0τ-λwt-ut-λe-λTut-Twtdt.Thus, on the one hand, (u(0),v(0),w(0)) ∈ ∂(R+3) ⇒ (u(τ),v(τ),w(τ)) ∈ ∂(R+3), and, on the other hand, (u(0),v(0),w(0)) ∈ int⁡(R+3) ⇒ (u(τ),v(τ),w(τ)) ∈ int⁡(R+3).Lemma 3 (see [4]).
Given(A,B)∈R+2 and ϕ a continuous and derivable function as there is t0≥0 verifying ϕ(t0)>0. Then, ∀t≥t0,(6)dϕdt≤ϕB-Aϕ⟹limsupt→+∞⁡ϕt≤BA,(7)dϕdt≥ϕB-Aϕ⟹liminft→+∞⁡ϕt≥BA.Theorem 4.
Let us assume thatα2>δ. Let us pose (8)Mu=4d2β1+β1d2+2α1β1d2+α12β1d24-β1if0<β1<4andMu=1+α1ifβ1=0,Mv=Mu+d2β2Mu+α2-δα2andmv=α2-δd2β2d2+α2,mu0=mu+β1mv,mu=-β1d2Mv2-α1d2Mv+1+α1,muε=mu0+mu0-ε2-4γkMuMv/d12,0≤ε≤mu0-2γkMuMvd1ormu0+2γkMuMvd1≤ε,Mw=kMuandmw=kmuε.(9)Ifmu0≥2γkMuMvd1,and, then, model 1.3 is permanent. Otherwise, the set defined by(10)A+=u,v,w∈R+3:muε≤u≤Mu,mv≤v≤Mvandmw≤w≤MwWhich is a bounded set, positively invariant for model 1.3.Proof.
Let us consider system1.3 and Lemma 3. (1)
One has the following:v˙(τ) does not depend on the variable w. So, one obtains the same results like those of the model with no delay, i.e.,liminft→+∞[v(t)]≥mv and limsupt→+∞vt≤Mv.(2)
One has the following:du/dt≤(1-u)u+(β1v+α1)(1-v/(u+d2))u because -γwv/(w+d1)<0.Then,du/dt≤(B1-A1u)u with A1,B1=1;1+α1 if β1=0and(A1,B1) = ((4-β1)/4; (4d2β1 + (β1d2+2α1)β1d2 + α12)/4β1d2) if 0<β1<4 and, in that case, maxv{(β1v+α1)(1 − v/(u+d2))u} = u[β1(u+d2)+α1]2/4β1(u+d2).By property (6) of Lemma 3, one obtains limsupt→+∞[u(t)] ≤ Mu.(3)
One knows that0≤u(τ). Let us suppose that u(0)>0 and, then, there exist t0 so that, ∀τ≥t0, 0<u(τ)≤Mu. Let us suppose that l = liminft→+∞[u(t)]. Thus, for any ε very tiny, there is t1>0 so that, ∀τ>t1, u(τ) ≥ l-ε>0.Let us poset2 = max⁡(t0,t1) and m=l-ε. Then, ∀τ>t2, 0<m < u(τ) ≤ Mu. Thus(11)∀τ>t2,0<wτ<uτwτm≤Muwτm.Hence,∀τ>t2, -γw(τ)/(w(τ)+d1) > -γ(uτwτ/m)/(uτwτ/m+d1) for g(x) = -γx/(x+d1) is decreasing upon R+. On the one hand,it is known thatw(t) = ∫t-Ttλe-λ(t-ξ)u(ξ)dξ and ∫-∞tλe-λ(t-ξ)dξ=1.Meanwhile,limsupt→+∞{∫t-Ttλe-λ(t-ξ)u(ξ)dξ} ≤ ∫t-Ttλe-λ(t-ξ)dξ × limsupt→+∞[u(t)] and, then,limsupτ→+∞[w(τ)] ≤ Mu(1-e-λT) = kMu=Mw. On the other hand,u˙/u = [1-u(τ)] + (β1v(τ)+α1)(1-vτ/uτ+d2) − (γwτ/wτ+d1)(vτ/uτ) and 1 + (β1v(τ)+α1)(1-vτ/uτ+d2) ≥ mu0 = -(β1/d2)Mv2-(α1/d2)Mv + 1 + α1 + β1mv. Then, τ>t2, and u˙ ≥ [mu0 − γw(τ)v(τ)/(u(τ)w(τ)+md1) − u]u(τ). So, du/dt≥(B2-A2u)u with A2=1 and B2 = mu(0) − γMvMw/md1. Therefore, applying property (7) of Lemma 3 one obtains liminft→+∞[u(t)] ≥ mu(0)-γMvMw/md1.Consequently,l ≥ mu(0)-γMvMw/(l-ε)d1 for m=l-ε. So, one obtains(12)l2-mu0+εl+γMuMwd1+mu0ε≥0.Ifmu(0)≥2γkMuMv/d1 then [mu(0)-ε]2 − 4γMwMv/d1 ≥ 0.Thus one obtainsliminft→+∞[u(t)] ≥ muε = (mu0+[mu0-ε]2-4γMwMv/d1)/2with0≤ε ≤ mu(0)-2γkMuMv/d1 or mu(0) + 2γkMuMv/d1 ≤ ε.(4)
It is known thatliminft→+∞∫t-Ttλe-λt-ξuξdξ ≥ ∫t-Ttλe-λ(t-ξ)dξ × liminft→+∞[u(t)]. So, liminft→+∞[w(t)] ≥ mw=kmuε.Remark 5.
Let us consider the notations of Theorem4. (1)
Let us consider the following assumptions:(13)α2>δ,mu0>0,d1mu024γMuMv<1andT<-1λln⁡1-d1mu024γMuMv.(14)α2>δ,mu0>0and0≤γ<d1mu024kMuMv.One remarks that each of assumptions (13) and (14) implies, respectively, condition (9).(2)
Ifmu(0)≥2γkMuMv/d1, one has 0<mu(0)/2≤muε≤liminft→+∞[u(t)]. So, we can adjust the minimum value of liminft→+∞[u(t)] by choosing the value of ε.


### 2.2. The Equilibria Points of Model1.3

#### 2.2.1. Points of Trivial Equilibria

If0<α2≤δ then system 1.3 admits two points of trivial equilibria: E0∗=(0;0;0),E1∗=(1+α1;0;k(1+α1)).Ifα2>δ≥0 then system 1.3 admits three points of trivial equilibria: E0∗=(0;0;0), E1∗=(1+α1;0;k(1+α1)), and E2∗=(0;(α2-δ)d2/α2;0).


#### 2.2.2. Points of Interior Equilibria

Givenp1(x)=a14x4+a13x3+a12x2+a11x+a10 such as(15)a14=-kβ22,a13=k-d1β22-2kβ2α2+kβ2β1δ-γβ2,a12=2β2α2k-d1-kα22+d1β22+kα1δβ2-kγβ2α2+β1β2δd1+kβ2d2+α2-δβ1δ-γβ2,a11=α1δβ2d1+kα2+k-d1α22+2β2α2d1+kβ1δ-γβ2d2α2-δ+β1δd1-kγα2β2d2+α2-δ,a10=d1α22+α1α2δd1+β1δd1-kγα2d2α2-δ.(1)
System1.3 does not admit any point of interior equilibria if β2Mu+α2-δ<0. (2)
Any interior equilibrium pointE3∗=(u∗;v∗,w∗) of system 1.3 verifies the following relations:(16)p1u∗=0,v∗=β2u∗+α2-δβ2u∗+α2u∗+d2withβ2u∗+α2-δ>0,w∗=ku∗.DesignatingPxy the projection over the plan xy, we study the border dynamics of the model.


## 2.1. Boundedness of the Solutions of Model1.3

Lemma 2.
The interiorint⁡(R+3) and the boundary ∂(R+3) of a positive cone R+3 are invariant for model 1.3.Proof.
Given0≤T0<+∞. Let τ be in [-T;T0[. One knows that (5)uτ=u0exp⁡∫0τ1-ut+β1vt+α11-vtut+d2-γvtwtutwt+d1dt,vτ=v0exp⁡∫0τβ2ut+α21-vtut+d2-δdt,wτ=w0exp⁡∫0τ-λwt-ut-λe-λTut-Twtdt.Thus, on the one hand, (u(0),v(0),w(0)) ∈ ∂(R+3) ⇒ (u(τ),v(τ),w(τ)) ∈ ∂(R+3), and, on the other hand, (u(0),v(0),w(0)) ∈ int⁡(R+3) ⇒ (u(τ),v(τ),w(τ)) ∈ int⁡(R+3).Lemma 3 (see [4]).
Given(A,B)∈R+2 and ϕ a continuous and derivable function as there is t0≥0 verifying ϕ(t0)>0. Then, ∀t≥t0,(6)dϕdt≤ϕB-Aϕ⟹limsupt→+∞⁡ϕt≤BA,(7)dϕdt≥ϕB-Aϕ⟹liminft→+∞⁡ϕt≥BA.Theorem 4.
Let us assume thatα2>δ. Let us pose (8)Mu=4d2β1+β1d2+2α1β1d2+α12β1d24-β1if0<β1<4andMu=1+α1ifβ1=0,Mv=Mu+d2β2Mu+α2-δα2andmv=α2-δd2β2d2+α2,mu0=mu+β1mv,mu=-β1d2Mv2-α1d2Mv+1+α1,muε=mu0+mu0-ε2-4γkMuMv/d12,0≤ε≤mu0-2γkMuMvd1ormu0+2γkMuMvd1≤ε,Mw=kMuandmw=kmuε.(9)Ifmu0≥2γkMuMvd1,and, then, model 1.3 is permanent. Otherwise, the set defined by(10)A+=u,v,w∈R+3:muε≤u≤Mu,mv≤v≤Mvandmw≤w≤MwWhich is a bounded set, positively invariant for model 1.3.Proof.
Let us consider system1.3 and Lemma 3. (1)
One has the following:v˙(τ) does not depend on the variable w. So, one obtains the same results like those of the model with no delay, i.e.,liminft→+∞[v(t)]≥mv and limsupt→+∞vt≤Mv.(2)
One has the following:du/dt≤(1-u)u+(β1v+α1)(1-v/(u+d2))u because -γwv/(w+d1)<0.Then,du/dt≤(B1-A1u)u with A1,B1=1;1+α1 if β1=0and(A1,B1) = ((4-β1)/4; (4d2β1 + (β1d2+2α1)β1d2 + α12)/4β1d2) if 0<β1<4 and, in that case, maxv{(β1v+α1)(1 − v/(u+d2))u} = u[β1(u+d2)+α1]2/4β1(u+d2).By property (6) of Lemma 3, one obtains limsupt→+∞[u(t)] ≤ Mu.(3)
One knows that0≤u(τ). Let us suppose that u(0)>0 and, then, there exist t0 so that, ∀τ≥t0, 0<u(τ)≤Mu. Let us suppose that l = liminft→+∞[u(t)]. Thus, for any ε very tiny, there is t1>0 so that, ∀τ>t1, u(τ) ≥ l-ε>0.Let us poset2 = max⁡(t0,t1) and m=l-ε. Then, ∀τ>t2, 0<m < u(τ) ≤ Mu. Thus(11)∀τ>t2,0<wτ<uτwτm≤Muwτm.Hence,∀τ>t2, -γw(τ)/(w(τ)+d1) > -γ(uτwτ/m)/(uτwτ/m+d1) for g(x) = -γx/(x+d1) is decreasing upon R+. On the one hand,it is known thatw(t) = ∫t-Ttλe-λ(t-ξ)u(ξ)dξ and ∫-∞tλe-λ(t-ξ)dξ=1.Meanwhile,limsupt→+∞{∫t-Ttλe-λ(t-ξ)u(ξ)dξ} ≤ ∫t-Ttλe-λ(t-ξ)dξ × limsupt→+∞[u(t)] and, then,limsupτ→+∞[w(τ)] ≤ Mu(1-e-λT) = kMu=Mw. On the other hand,u˙/u = [1-u(τ)] + (β1v(τ)+α1)(1-vτ/uτ+d2) − (γwτ/wτ+d1)(vτ/uτ) and 1 + (β1v(τ)+α1)(1-vτ/uτ+d2) ≥ mu0 = -(β1/d2)Mv2-(α1/d2)Mv + 1 + α1 + β1mv. Then, τ>t2, and u˙ ≥ [mu0 − γw(τ)v(τ)/(u(τ)w(τ)+md1) − u]u(τ). So, du/dt≥(B2-A2u)u with A2=1 and B2 = mu(0) − γMvMw/md1. Therefore, applying property (7) of Lemma 3 one obtains liminft→+∞[u(t)] ≥ mu(0)-γMvMw/md1.Consequently,l ≥ mu(0)-γMvMw/(l-ε)d1 for m=l-ε. So, one obtains(12)l2-mu0+εl+γMuMwd1+mu0ε≥0.Ifmu(0)≥2γkMuMv/d1 then [mu(0)-ε]2 − 4γMwMv/d1 ≥ 0.Thus one obtainsliminft→+∞[u(t)] ≥ muε = (mu0+[mu0-ε]2-4γMwMv/d1)/2with0≤ε ≤ mu(0)-2γkMuMv/d1 or mu(0) + 2γkMuMv/d1 ≤ ε.(4)
It is known thatliminft→+∞∫t-Ttλe-λt-ξuξdξ ≥ ∫t-Ttλe-λ(t-ξ)dξ × liminft→+∞[u(t)]. So, liminft→+∞[w(t)] ≥ mw=kmuε.Remark 5.
Let us consider the notations of Theorem4. (1)
Let us consider the following assumptions:(13)α2>δ,mu0>0,d1mu024γMuMv<1andT<-1λln⁡1-d1mu024γMuMv.(14)α2>δ,mu0>0and0≤γ<d1mu024kMuMv.One remarks that each of assumptions (13) and (14) implies, respectively, condition (9).(2)
Ifmu(0)≥2γkMuMv/d1, one has 0<mu(0)/2≤muε≤liminft→+∞[u(t)]. So, we can adjust the minimum value of liminft→+∞[u(t)] by choosing the value of ε.


## 2.2. The Equilibria Points of Model1.3

### 2.2.1. Points of Trivial Equilibria

If0<α2≤δ then system 1.3 admits two points of trivial equilibria: E0∗=(0;0;0),E1∗=(1+α1;0;k(1+α1)).Ifα2>δ≥0 then system 1.3 admits three points of trivial equilibria: E0∗=(0;0;0), E1∗=(1+α1;0;k(1+α1)), and E2∗=(0;(α2-δ)d2/α2;0).


### 2.2.2. Points of Interior Equilibria

Givenp1(x)=a14x4+a13x3+a12x2+a11x+a10 such as(15)a14=-kβ22,a13=k-d1β22-2kβ2α2+kβ2β1δ-γβ2,a12=2β2α2k-d1-kα22+d1β22+kα1δβ2-kγβ2α2+β1β2δd1+kβ2d2+α2-δβ1δ-γβ2,a11=α1δβ2d1+kα2+k-d1α22+2β2α2d1+kβ1δ-γβ2d2α2-δ+β1δd1-kγα2β2d2+α2-δ,a10=d1α22+α1α2δd1+β1δd1-kγα2d2α2-δ.(1)
System1.3 does not admit any point of interior equilibria if β2Mu+α2-δ<0. (2)
Any interior equilibrium pointE3∗=(u∗;v∗,w∗) of system 1.3 verifies the following relations:(16)p1u∗=0,v∗=β2u∗+α2-δβ2u∗+α2u∗+d2withβ2u∗+α2-δ>0,w∗=ku∗.DesignatingPxy the projection over the plan xy, we study the border dynamics of the model.


## 2.2.1. Points of Trivial Equilibria

If0<α2≤δ then system 1.3 admits two points of trivial equilibria: E0∗=(0;0;0),E1∗=(1+α1;0;k(1+α1)).Ifα2>δ≥0 then system 1.3 admits three points of trivial equilibria: E0∗=(0;0;0), E1∗=(1+α1;0;k(1+α1)), and E2∗=(0;(α2-δ)d2/α2;0).


## 2.2.2. Points of Interior Equilibria

Givenp1(x)=a14x4+a13x3+a12x2+a11x+a10 such as(15)a14=-kβ22,a13=k-d1β22-2kβ2α2+kβ2β1δ-γβ2,a12=2β2α2k-d1-kα22+d1β22+kα1δβ2-kγβ2α2+β1β2δd1+kβ2d2+α2-δβ1δ-γβ2,a11=α1δβ2d1+kα2+k-d1α22+2β2α2d1+kβ1δ-γβ2d2α2-δ+β1δd1-kγα2β2d2+α2-δ,a10=d1α22+α1α2δd1+β1δd1-kγα2d2α2-δ.(1)
System1.3 does not admit any point of interior equilibria if β2Mu+α2-δ<0. (2)
Any interior equilibrium pointE3∗=(u∗;v∗,w∗) of system 1.3 verifies the following relations:(16)p1u∗=0,v∗=β2u∗+α2-δβ2u∗+α2u∗+d2withβ2u∗+α2-δ>0,w∗=ku∗.DesignatingPxy the projection over the plan xy, we study the border dynamics of the model.


## 3. Border Dynamics on the Planuv

Onuv plan, model 1.3 becomes 3.1u˙τ=1-uτuτ+β1vτ+α11-vτuτ+d2uτ,v˙τ=β2uτ+α21-vτuτ+d2vτ-δv,u,v,w∈C10;+∞;R+,u0>0,v0>0,u0=φ∈C-T;0;R+,with(d2,α1,α2,β1,β2,δ,T,λ)∈(R+∗)7.The equilibria points of model3.1 are(17)U01=PuvE0∗=0,0,U11=PuvE1∗=1+α1;0,(18)U21=PuvE2∗=0;α2-δd2α2,(19)U31=u¯;β2u¯+α2-δu¯+d2β2u¯+α2ifu¯>0andβ2u¯+α2-δ>0,(20)wherep0u¯=a03u¯3+a02u¯2+a01u¯+a00=0.(21)Witha03=-β22,a02=β2-2α2+δβ1β2,a01=α22β2-α2+β1δβ2d2+α2-δ+δβ2α1,a00=α22+δβ1d2α2-δ+α1α2δ.Let us give below the results on the permanence of the model. Therefore, let us consider the notations of Theorem 4. If α2>δ and mu(0)>0 then model 3.1 is permanent. Among others, the set defined by(22)A0=u,v∈R+2:mu0≤u≤Mu,andmv≤v≤Mvwhich is a bounded set, positively invariant for model 3.1.


### 3.1. Local Stability of the Equilibria Points of Model3.1

Performing the spectral study of Jacobian matrix of the system linearized around each of the points of equilibrium, one obtains, classically, the following conclusions:(1)
Stability ofU0(1)=(0;0):
(a)
U0(1) is an unstable node if α2>δ.(b)
U0(1) is a point unstable saddle repulsive following direction u and attractive following direction v if α2<δ.(2)
StabilityU1(1)=(1+α1;0):
(a)
U1(1) is stable if α2+(1+α1)β2<δ.(b)
U1(1) is a point unstable saddle attractive following direction u and repulsive following direction w1 if α2+(1+α1)β2>δ.(3)
Stability ofU2(1)=(0;(α2-δ)d2/α2) for α2>δ:
(a)
U2(1) is stable if ψ(δ)<0.(b)
U2(1) is a point of unstable saddle if ψ(δ)>0, repulsive following w2, and attractive following v. Consider(23)ψx=-d2β1x2+d2β1α2+α1α2x+α22.(4)
GivenU3(1)=(u¯;v¯) interior equilibrium point of 3.1 testifying the system (19)-(20) and J its associated Jacobian matrix.
(a)
U3(1) is stable (stable node or stable focus) if det⁡(J)>0 and Tr(J)<0.(b)
U3(1) is marginal or center if det⁡(J)≥0 and Tr(J)=0.(c)
U3(1) is unstable if det⁡(J)<0 or (det⁡(J)>0 and Tr(J)>0), precisely:
(i)
U3(1) is a node or a focus if det⁡(J)>0 and Tr(J)>0,(ii)
U3(1) is a point unstable saddle if det⁡(J)<0.WithTr(J) the trace and det⁡(J) determinant of J, vectors (24)w1=1+α1β1-α11+α1+d2;1+α1β2+1+α2-δ.and w2=ψ(δ)/α22+α2-δ;[β2d2δ+α2(α2-δ)](α2-δ)/α22.


### 3.2. Global Stability ofU3(1)=(u¯;v¯)

Now let us define the conditions for which the stability of the product and the stock of capital of the economy is global; i.e., it does not depend on the produced quantities and level of stock at the initial moment. For this study, we define an appropriate Lyapunov function.Theorem 6.
Let us consider the following assumptions:(25)(3.1)admitsauniqueinteriorequilibriumpointU31=u¯;v¯,(26)0<β1<4,α2>δ,β2d2<2α2,d2<d1,2δβ1α2<α1Mu+d2,(27)β1Mvd2+α1d2<β1d22α2-δα2Mu+d2+δβ11+d2β2/2α2α2,(28)1+α1d2-1+2β1d22α12<0.where Mu and Mv are defined in Theorem 4.
If assumptions (25), (26), (27), (28), (29), (30), (31), (35), (36), and (37) are verified, then, the unique point of the interior equilibrium of model 3.1 is globally and asymptotically stable.Proof.
Let us consider system3.1. Let us suppose that assumption (35) is verified and, then, model 3.1 admits a unique interior equilibrium point U3(1)=(u¯;v¯) verifying(29)1=u¯-α11-v¯u¯+d2,α2-δ=-β2u¯+β2u¯+α2v¯u¯+d2.Let us note that λ=β1d2/2α2. Given V1:R2→R and V2:R2→R one has(30)V1u,v=u-u¯-u¯ln⁡uu¯=∫u¯u1-u¯xdx,(31)V2u,v=λv-v¯-v¯ln⁡vv¯=λ∫v¯v1-v¯xdx.Considering the Lyapunov functional V:R2→R one has(32)Vu,v=V1u,v+V2u,v.Let us pose (33)gu,v=-1+β1v2+α1vu¯+d2u+d2;hu,v=12β1-β1v+v¯+α1u¯+d2+λ2β2-β2v¯u¯+d2+β2u+α2vu¯+d2u+d2.By using relation (29) and model 3.1, one obtains (34)dVdt≤ϕ1u,vu-u¯2+ϕ2u,vv-v¯2with ϕ1=g(u,v)+hu,v and ϕ2=hu,v-λ(β2u+α2)/(u¯+d2), ∀(u,v)∈R2.
Let us overestimateg(u,v) and h(u,v) and, then, ϕ1(u,v) and ϕ2(u,v).(1)
One hasg(u,v)<R0<0, ∀(u,v)∈R2, if R0=-1+(β1Mv2+α1Mv)/d22<0.(2)
ifβ2<2α2/d2 then ∂h/∂v<0. So, h(Mu,Mv)≤h(u,v)≤h(0,0). Therefore, h(0,0)<0 if β2d2<2α2,d2<d1,2δβ1/α2<α1/(Mu+d2). Consequently, hu,v = -hu,v < -h(Mu,Mv) < R1=β1Mv/2d2-(d2β1Mv/2α2)β2Mu+α2/(Mu+d2)2+(1/2)[-(β1δ/α2)(1+β2d2/2α2)+α1/d2]. Thus, ϕ1(u,v)<R0+R1 and ϕ2(u,v)<R1-β1d2/2(Mu+d2) if β2d2<2α2, d2<d1, and 2δβ1/α2<α1/(Mu+d2).Hence,dV/dt<0, ∀(u,v)∈R2, if(35)0<β1<4,α2>δ,β2<α2d2,2δβ1α2<α1Mu+d2,(36)β1Mv2d2+α12d2<β1d22Mu+d2+d2β1Mv2α2β2Mu+α2Mu+d22+β1δ2α21+β2d22α2,(37)andβ1Mv2+α1Mvd22-1+β1d22Mu+d2<0.The model is permanent ifmu=-(β1/d2)Mv2-(α1/d2)Mv+1+α1>0. Then, (β1/d22)Mv2+(α1/d22)Mv<(1+α1)/d2. Moreover, 1/(Mu+d2)<4d2/α12.So, from formulas (35), one obtains (27) ⇒ (36) and (28) ⇒ (37).
Hence,dV/dt<0, ∀(u,v)∈R2, if assumptions (25)–(28) are verified.Remark 7.
Asβ1=0, the unique interior equilibrium point of model 3.1 is globally and asymptotically stable if the following assumptions are verified:(38)α2>δ,β2<α2d2,δβ2α2<α1Mu+d2,(39)-1+1+α1d2+4α2d2α12<0,(40)α12d2+δβ2α2<3α2-δ2Mu+d2,
with constantMu and Mv defined in Theorem 4.
In fact, it is worth taking the Lyapunov functionalV:R2→R so that V(u,v)=V1(u,v)+V2(u,v) with V1:R2→R and V2:R2→R so that(41)V1u,v=u-u¯-u¯ln⁡uu¯=∫u¯u1-u¯xdx,(42)V2u,v=v-v¯-v¯ln⁡vv¯=∫v¯v1-v¯xdx.Considering (38)–(40), one shows, similarly in β1≠0, that dV/dt<0, ∀(u,v)∈R2.


## 3.1. Local Stability of the Equilibria Points of Model3.1

Performing the spectral study of Jacobian matrix of the system linearized around each of the points of equilibrium, one obtains, classically, the following conclusions:(1)
Stability ofU0(1)=(0;0):
(a)
U0(1) is an unstable node if α2>δ.(b)
U0(1) is a point unstable saddle repulsive following direction u and attractive following direction v if α2<δ.(2)
StabilityU1(1)=(1+α1;0):
(a)
U1(1) is stable if α2+(1+α1)β2<δ.(b)
U1(1) is a point unstable saddle attractive following direction u and repulsive following direction w1 if α2+(1+α1)β2>δ.(3)
Stability ofU2(1)=(0;(α2-δ)d2/α2) for α2>δ:
(a)
U2(1) is stable if ψ(δ)<0.(b)
U2(1) is a point of unstable saddle if ψ(δ)>0, repulsive following w2, and attractive following v. Consider(23)ψx=-d2β1x2+d2β1α2+α1α2x+α22.(4)
GivenU3(1)=(u¯;v¯) interior equilibrium point of 3.1 testifying the system (19)-(20) and J its associated Jacobian matrix.
(a)
U3(1) is stable (stable node or stable focus) if det⁡(J)>0 and Tr(J)<0.(b)
U3(1) is marginal or center if det⁡(J)≥0 and Tr(J)=0.(c)
U3(1) is unstable if det⁡(J)<0 or (det⁡(J)>0 and Tr(J)>0), precisely:
(i)
U3(1) is a node or a focus if det⁡(J)>0 and Tr(J)>0,(ii)
U3(1) is a point unstable saddle if det⁡(J)<0.WithTr(J) the trace and det⁡(J) determinant of J, vectors (24)w1=1+α1β1-α11+α1+d2;1+α1β2+1+α2-δ.and w2=ψ(δ)/α22+α2-δ;[β2d2δ+α2(α2-δ)](α2-δ)/α22.


## 3.2. Global Stability ofU3(1)=(u¯;v¯)

Now let us define the conditions for which the stability of the product and the stock of capital of the economy is global; i.e., it does not depend on the produced quantities and level of stock at the initial moment. For this study, we define an appropriate Lyapunov function.Theorem 6.
Let us consider the following assumptions:(25)(3.1)admitsauniqueinteriorequilibriumpointU31=u¯;v¯,(26)0<β1<4,α2>δ,β2d2<2α2,d2<d1,2δβ1α2<α1Mu+d2,(27)β1Mvd2+α1d2<β1d22α2-δα2Mu+d2+δβ11+d2β2/2α2α2,(28)1+α1d2-1+2β1d22α12<0.where Mu and Mv are defined in Theorem 4.
If assumptions (25), (26), (27), (28), (29), (30), (31), (35), (36), and (37) are verified, then, the unique point of the interior equilibrium of model 3.1 is globally and asymptotically stable.Proof.
Let us consider system3.1. Let us suppose that assumption (35) is verified and, then, model 3.1 admits a unique interior equilibrium point U3(1)=(u¯;v¯) verifying(29)1=u¯-α11-v¯u¯+d2,α2-δ=-β2u¯+β2u¯+α2v¯u¯+d2.Let us note that λ=β1d2/2α2. Given V1:R2→R and V2:R2→R one has(30)V1u,v=u-u¯-u¯ln⁡uu¯=∫u¯u1-u¯xdx,(31)V2u,v=λv-v¯-v¯ln⁡vv¯=λ∫v¯v1-v¯xdx.Considering the Lyapunov functional V:R2→R one has(32)Vu,v=V1u,v+V2u,v.Let us pose (33)gu,v=-1+β1v2+α1vu¯+d2u+d2;hu,v=12β1-β1v+v¯+α1u¯+d2+λ2β2-β2v¯u¯+d2+β2u+α2vu¯+d2u+d2.By using relation (29) and model 3.1, one obtains (34)dVdt≤ϕ1u,vu-u¯2+ϕ2u,vv-v¯2with ϕ1=g(u,v)+hu,v and ϕ2=hu,v-λ(β2u+α2)/(u¯+d2), ∀(u,v)∈R2.
Let us overestimateg(u,v) and h(u,v) and, then, ϕ1(u,v) and ϕ2(u,v).(1)
One hasg(u,v)<R0<0, ∀(u,v)∈R2, if R0=-1+(β1Mv2+α1Mv)/d22<0.(2)
ifβ2<2α2/d2 then ∂h/∂v<0. So, h(Mu,Mv)≤h(u,v)≤h(0,0). Therefore, h(0,0)<0 if β2d2<2α2,d2<d1,2δβ1/α2<α1/(Mu+d2). Consequently, hu,v = -hu,v < -h(Mu,Mv) < R1=β1Mv/2d2-(d2β1Mv/2α2)β2Mu+α2/(Mu+d2)2+(1/2)[-(β1δ/α2)(1+β2d2/2α2)+α1/d2]. Thus, ϕ1(u,v)<R0+R1 and ϕ2(u,v)<R1-β1d2/2(Mu+d2) if β2d2<2α2, d2<d1, and 2δβ1/α2<α1/(Mu+d2).Hence,dV/dt<0, ∀(u,v)∈R2, if(35)0<β1<4,α2>δ,β2<α2d2,2δβ1α2<α1Mu+d2,(36)β1Mv2d2+α12d2<β1d22Mu+d2+d2β1Mv2α2β2Mu+α2Mu+d22+β1δ2α21+β2d22α2,(37)andβ1Mv2+α1Mvd22-1+β1d22Mu+d2<0.The model is permanent ifmu=-(β1/d2)Mv2-(α1/d2)Mv+1+α1>0. Then, (β1/d22)Mv2+(α1/d22)Mv<(1+α1)/d2. Moreover, 1/(Mu+d2)<4d2/α12.So, from formulas (35), one obtains (27) ⇒ (36) and (28) ⇒ (37).
Hence,dV/dt<0, ∀(u,v)∈R2, if assumptions (25)–(28) are verified.Remark 7.
Asβ1=0, the unique interior equilibrium point of model 3.1 is globally and asymptotically stable if the following assumptions are verified:(38)α2>δ,β2<α2d2,δβ2α2<α1Mu+d2,(39)-1+1+α1d2+4α2d2α12<0,(40)α12d2+δβ2α2<3α2-δ2Mu+d2,
with constantMu and Mv defined in Theorem 4.
In fact, it is worth taking the Lyapunov functionalV:R2→R so that V(u,v)=V1(u,v)+V2(u,v) with V1:R2→R and V2:R2→R so that(41)V1u,v=u-u¯-u¯ln⁡uu¯=∫u¯u1-u¯xdx,(42)V2u,v=v-v¯-v¯ln⁡vv¯=∫v¯v1-v¯xdx.Considering (38)–(40), one shows, similarly in β1≠0, that dV/dt<0, ∀(u,v)∈R2.


## 4. Border Dynamics of Planuw

On the planuw model 1.3 becomes 4.1u˙τ=1+α1-uτuτ,w˙=-λw-u-λe-λTuτ-T,u,w∈C10;+∞​;R+,u0>0,u0=φ∈C-T;0;R+with(α1,T,λ)∈(R+∗)3. The equilibria of model 4.1 are(43)U02=PuwE0∗=0,0,U12=PuwE1∗=1+α1;k1+α1.Let Mu(2)=1+α1, 0<mu(2)<1+α1, Mw(2)=k(1+α1), and mw(2)=kmu(2). The model is permanent and the set defined by(44)A2=u,w∈R+2:mu2≤u≤Mu2,andmw2≤w≤Mw2is limited, positively invariant.Concerning the local stability, one obtains, for any delayT, the following results:(1)
U0(2)=(0;0) is an unstable saddle point, repulsive along the direction u, and attractive along the direction w.(2)
U1(2)=(1+α1;k(1+α1)) is stable.The global stability of equilibriumU1(2) is given in the following theorem.Theorem 8.
(45) I f   T < ln ⁡ 2 λ and λ < 1 + α 1 1 - ln ⁡ 2and, then, the unique interior equilibrium point U1(2)=(1+α1;k(1+α1)) of model 4.1 is globally and asymptotically stable.Proof.
Let us considerU1(2)=(u¯,w¯) with u¯=1+α1 and w¯=k(1+α1). Let us pose
U = l n ( u / u ¯ ) and W=ln(w/w¯). Then, one obtains the following system:(46)U˙τ=-u¯eUτ-1,W˙τ=λe-wτkeUτ-1-keWτ-1-e-λTeUτ-T-1.Let us note that V1(τ)=Uτ.
Then, the superior derivativeD+V1(τ) of V1(τ) in relation to time, alongside the solutions of (46), gives(47)D+V1τ≤-u¯eUτ-1=-u-u¯.Let us note that V21(τ)=Wτ. Considering (u,w)∈A(2), one has mu(2)/u¯≤eU(τ)≤Mu(2)/u¯ and mw(2)/w¯≤eW(τ)≤Mw(2)/w¯. Hence, the superior derivative of V21(τ) in relation to time, alongside solutions (46), gives(48)D+V21τ≤λeUτ-1-λw∗mw2eWτ-1+Mu2λe-λTk∫τ-TτeUs-1ds.Considering the functional: V22(τ)=(Mu2λe-λT/k)∫τ-Tτ∫vτeUs-1dsdv then(49)D+V22τ≤TMu2λe-λTkeUτ-1-Mu2λe-λTk∫τ-TτeUs-1ds.Let us pose V2(τ)=V21(τ)+V22(τ). Thus,(50)D+V2τ≤λ1+TMu2e-λTkeUτ-1-λw∗mw2eWτ-1.(51)Then,D+V2τ≤λu¯1+TMu2e-λTku-u¯-λmw2w-w¯.Let V(τ)=V1(τ)+V2(τ). Then,(52)D+Vτ≤-1-λu¯1+TMu2e-λTku-u¯-λmw2w-w¯.So, D+V(τ)<0 if (1+(Te-λT/1-e-λT)Mu(2)<(1+α1)/λ). Consequently, D+V(τ)<0 if (T<(ln⁡2)/λ and λ<(1+α1)(1-ln⁡2).


## 5. Border Dynamics of the Planvw

On thevw plan model 1.3 becomes 5.1v˙τ=α2-δ-α2vτd2vτ,w˙=-λw-λe-λT10;Tτuτ-T,v,w∈C10;+∞​;R+u0>0,u0=φ∈C-T;0;R+with(d2,α2,δ,T,λ)∈(R+∗)4 and 10;T(τ)=1if0≤τ≤T,10;T(τ)=0ifτ>T.The equilibria of model5.1 are as follows: for τ≥T,(53)U03=PvwU0=0,0,U23=PvwU2=α2-δd2α2;0.Model 4.1 is nonpermanent. In fact, limt→+∞wt=0.Let us now give the results upon stability of the equilibria of model5.1.(1)
The equilibriumU0(3)=(0;0) isstable if α2<δ and a pointunstable saddle if α2>δ.(2)
The equilibriumU2(3)=((α2-δ)d2/α2;0) for α2>δ is stable.Moreover, the equilibriumU2(3)=((α2-δ)d2/α2;0) is globally and asymptotically stable. In effect, limt→+∞vt=(α2-δ)d2/α2 and limt→+∞wt=0.


## 6. Local Stability of Model1.3

Let us consider model1.3.Let us pose thatUi(τ)=u(τ)-ui∗, Vi(τ)=v(τ)-vi∗, and Wi(τ)=w(τ)-wi∗ for i=0,1,2,3 with(54)u0∗=0,u1∗=1+α1,u2∗=0,u3∗=u∗(55)v0∗=0,v1∗=0,v2∗=α2-δd2α2,v3∗=v∗(56)w0∗=0,w1∗=k1+α1,w2∗=0,w3∗=ku∗.The system linearized around the point of equilibrium Ei∗=(ui∗,vi∗,wi∗),i=0,1,2,3, is(57)U˙iτ=A11iUiτ+A12iViτ+A13iWiτ,V˙iτ=A21iUiτ+A22iViτ,W˙iτ=λUiτ-λWiτ-λe-λTUiτ-T,with(58)A11i=1-2ui∗+β1vi∗+α11-vi∗ui∗+d2+β1vi∗+α1ui∗vi∗ui∗+d22,(59)A12i=β1-2β1ui∗+d2vi∗-α1ui∗+d2ui∗-γwi∗wi∗+d1,(60)A13i=-γd1vi∗wi∗+d12,A21i=β21-vi∗ui∗+d2+β2ui∗+α2vi∗ui∗+d22vi∗,(61)A22i=β2ui∗+α21-2vi∗ui∗+d2-δ,A23i=0,(62)A31i=λ,A32i=0,A33i=-λ,A34i=-λe-λT.The characteristic equation of (57) is(63)Δix,T=Pix,T+e-xTQix,T=0(64)withPix,T=-x3+aix2+bix+ci,Qix,T=e-λTDix+Ei(65)where,Di=-λA13i,Ei=λA13iA22i,ai=-A11i+A22i-λ,(66)bi=-λA11i+A22i+A13i+A21iA12i-A11iA22i,(67)ci=λA11iA22i-A21iA12i+A13iA22i.


### 6.1. Local Stability ofE0∗ and E1∗

We are in situations whereQi(x,T)=0. Let us use the following criterion Routh-Hurwitz: the interior equilibrium Ei∗ is stable if a(i)>0, b(i)>0, c(i)>0, and a(i)b(i)-c(i)>0 and unstable if not.Theorem 9.
Let us suppose thatα2>δ. Then, the equilibria E0∗ and E1∗ of model 1.3 are all unstable.Proof.
Let us consider formula (63)–(67). (1)
The equation ofE0∗ is Δ0(x,T)=x3+a(0)x2+b(0)x+c(0)=0.One has the following:a(0)=λ-(1+α1+α2-δ) and b(0)=(1+α1)(α2-δ)-λ(1+α1+α2-δ). Let us suppose that (a(0)>0 and b(0)>0). Then, one would have (1+α1+α2-δ)<λ<(1+α1)(α2-δ)/(1+α1+α2-δ). Consequently, (1+α1)2+(α2-δ)2+(1+α1)(α2-δ)<0 which is absurd. Hence, the equilibrium E0∗ of model 1.3 is unstable by application of the Routh-Hurwitz criterion.(2)
One hasΔ1(x,T)=x3+a(1)x2+b(1)x+c(1)=0. Meanwhile, α2>δ and, then, c(1)=-λ(1+α1)[β2(1+α1)+α2-δ]<0 so the equilibrium E0∗ of model 1.3 is unstable by application of Routh-Hurwitz criterion.


### 6.2. Local Stability ofE2∗ and E3∗

We are in the situation whereQi(x,T)≠0. In order to assess the influence of delay T over the local stability of model 1.3, we use the results upon the local stability of model obtained taking T=0 into model 1.3 and the results of [5] upon the local stability of the delay systems. Let us consider the characteristic equation (63).(i)
One has the following:Qi(z,T)=0⇒z=-E(i)/D(i)∈R. Then, Pi(x,T) and Qi(x,T) do not have any common imaginary roots.(ii)
One has the following:Pi(x,T) and Qi(x,T) are polynomials with real coefficientssoPi(-iω,T)¯=Pi(iω,T) and Qi(-iω,T)¯=Qi(iω,T).(iii)
If-(1/λ)ln⁡c(i)/E(i)≠T so Pi(0,T)+Qi(0,T)=E(i)e-λT-c(i)≠0.(iv)
Let us posex=ρeiφ. Then, Pi(x,T)/Qi(x,T)~1/ρ4 when ρ→+∞.(68)Thuslimsupx→+∞Qix,TPix,T=0<1.Let us consider the function defined onR by the following: Fi(ω)=Piiω,T2-Qiiω,T2.(69)Then,Fω=biω-ω32+ci-aiω22-e-2λTDiω2+Ei(70)SoFiω=ω6+m2ω4+m1ω2+m0,where(71)m2=ai2-2bi,m1=bi2-2aici-e-2TλDi,m0=ci2-Eie-2λT.Let us determineδ(Ti∗)=sgn⁡(dRe⁡(x)/dT)x=iω(Ti∗) the sign of the real part of a solution x of the characteristic equation Δi(x,T)=0.Lemma 10.
Let us considerx, a solution of the characteristic equation Δi(x,T)=Pi(x,T)+e-xTQi(x,T)=0, with (72)Pix,T=-x3+aix2+bix+ciandQix,T=e-λTDix+Ei.Given ωi=ω(Ti∗) positive root of Fi(ω) and Ti∗ the associated delay certifying the following relation:(73)Ti∗=1ωiarctan⁡aiDi-Eiω3+biEi-ciDiω-Diω4+-aiEi+biDiω2+ciEi+2nπωi,n∈N.Let us pose δ(Ti∗)=sgn⁡(dRe⁡(x)/dT)x=iω(Ti∗). Then,(74)δTi∗=sgn⁡Λ1iH3i+Λ2iH1iH2iwith (75)H1i=λ2+ω2,H2i=λEi-Diω22+Ei+λDi2ω2,H3i=ci-aiω22+bi-ω2ω2,Λ1i=-Ti∗Di2-λ2Di2ω4+λDiEi-Ti∗Ei-λDi2-2λDiEi-λ2Di2ω2+λ3DiEi-λ2Ei2Ti∗,Λ2i=-3aiω4+2ω3+3ci+aibi-2biω2-bici.Proof.
Givenωi=ω(Ti∗) positive root of Fi(ω). Let us assume that there is a delay Ti∗ so that(76)Ti∗=1ωTi∗arctan⁡-Im⁡Piiω,Ti∗/Qiiω,Ti∗Re⁡Piiω,Ti∗/Qiiω,Ti∗+2nπωTi∗,n∈N.(77)Then,Ti∗=1ωiarctan⁡aiDi-Eiω3+biEi-ciDiω-Diω4+-aiEi+biDiω2+ciEi+2nπωi,n∈N.Let us pose(78)δTi∗=sgn⁡dRe⁡xdTx=iωTi∗=sgn⁡dRe⁡xdTx=iωTi∗-1.One has (79)Δix,T=Pix,T+e-xTQix,T=0⟹dPix,TdT-TdxdT+xe-xTQix,T+e-xTdQix,TdT=0.Therefore, Pix,T=-x3+aix2+bix+ci and Qix,T=e-λTDix+Ei. Thus, e-xTDie-λT-TQix,T-3x2+2aix+bi(dx/dT)-e-xTxQi(x,T)+λe-λT(D(i)x+E(i))=0.(80)Hence,dxdT-1=e-xTDie-λT-TQix,T-3x2+2aix+bie-xTxQix,T+λe-λTDix+Ei.Thus,dxdT-1=-Tx+λ+Dix+λDix+Ei-3x2+2aix+bix3+aix2+bix+ciand, meanwhile, δ(Ti∗)=sgn⁡(dRe⁡x/dT)x=iω(Ti∗)-1. Then, δTi∗=sgn⁡Λ1iH3i+Λ2iH1iH2i
withH1i=λ2+ω2, H2i = (λEi-Diω2)2 + (Ei+λDi)2ω2, H3i = (ci-aiω2)2 + (bi-ω)2ω2, Λ1i = -Ti∗Di2-λ2Di2ω4 + λDiEi − Ti∗Ei-λDi2,-2λDiEi − λ2Di2ω2 + λ3DiEi-λ2Ei2Ti∗, and Λ2i = -3a(i)ω4 + 2ω3 + (3c(i)+a(i)b(i)-2b(i))ω2 − b(i)c(i).It is noticed that the analysis of the local stability of the system with delay whenQi(x,T)≠0 depends on the existence of positive root for function Fi(ω). The coefficients m0 and m1, of Fi(ω), depend on T. Therefore, its positive root ω depends on T. Using the Cartan method and Viet formulas, one proves on the one hand that Fi(ω) admits at least a positive root if m0<0. On the other hand, if m0≥0, then, Fi(ω) admits either of two positive roots whereas Fi(ω) does not admit any positive root.The conditionm0<0 is equivalent to the following assumption:(81)0≤T≤Ts=-12λln⁡ci2EiwithEi>0andci2<Ei.Let us denotedet⁡Ji=A11(i)A22(i)-A21(i)A12(i), p=-m1m3+m22/3, and q=-m0m32-2(m2/3)3+m1m2m3/3. Then, one obtains the following proposition.Proposition 11.
Let us suppose thatA13(i)A22(i)>0.
Let us poseλmax(i)=A13(i)A22(i)/det⁡Ji+A13(i)A22(i)2 and Ts(i)=-1/2λln⁡(ci2/E(i)), i=2,3.(1)
Fi(ω) admits at least a positive root if(82)0≤λ<λmaxiand0≤T<Tsi.(2)
Fi(ω) does not admit any positive root if one of the following conditions is verified:  (83)Tsi≤Tand27q2-4p3>0.(84)Tsi≤T,27q2-4p3=0andm1m2-m0>0.(85)Tsi≤T,27q2-4p3<0,m2<0,m1<0andm1m2-m0>0.Now, let us use the results of Theorem4 from [5] to study the stability of these points of equilibriums E2∗ and E3∗.Theorem 12.
One poses(86)x-=β1d2+α1+γd2/d12-β1d2+α1+γd2/d122-4β1d2γd2/d12-12β1d2.Let us consider one of these assumptions (82)–(85) and one of the following conditions:(87)γd2d12<1,(88)γd2d12≥1andmax⁡0;x-≤δα2<1,(89)γd2d12≥1,0≤δα2<min⁡1;x-andT≠-1λln⁡1-d12ψδγα2α2-δ,(90)det⁡J=A113A223-A213A123>0andTrJ=A113+A223<0.With ψ(δ) defined in formula (23), consider the following. (1)
Let us suppose that one of assumptions (83)–(85) is verified for i=2,3. Then, there is no change of stability for equilibrium Ei∗,i=2,3.(2)
Let us suppose that assumptions (82) and one of these assumptions (87)–(89) are verified for i=2.
(a)
Ifδ(T2∗)>0 then the equilibrium E2∗ is unstable for any T∈0;Ts.(b)
Ifδ(T2∗)<0 then the equilibrium E2∗ is unstable if 0≤T<T2∗ and stable if T2∗≤T.(3)
Let us suppose that assumption (82) is verified for i=3.
(a)
In caseδ(T3∗)>0,
(i)
If assumption (90) is verified then the equilibrium E3∗ is stable if 0≤T<T3∗ and unstable if T3∗≤T.(ii)
If assumption (90) is not verified then the equilibrium E3∗ is unstable for any T∈0;Ts.(b)
Consideringδ(T3∗)<0,
(i)
If assumption (90) is verified then the equilibrium E3∗ is stable for every T∈0;Ts.(ii)
If assumption (90) is not verified then the equilibrium E3∗ is unstable if 0≤T<T3∗ and stable if T3∗≤T.
WithTi∗ and δ(Ti∗), i=2,3 are the denotations defined in Lemma 10.Proof.
Let us consider formula (63)–(67). Let us consider the function Fi(ω) defined in relation to the formula (70)-(71) for any point of equilibrium Ei∗, i=2,3. Let us consider assumptions (82). Then, Ts(i) exists so that, for any T≤Ts(i), Fi(ω) admits ωi=ω(Ti∗) a positive root (see Proposition 11). Considering Ti∗ the associated delay determined from Lemma 10, consider the following:(1)
Let us suppose that one of assumptions (83)–(85) is verified in i=2,3. Then, ∀T>Ts(i), Fi(ω) does not admit any positive root. Thus, there is no change of stability of the equilibrium Ei∗,i=2,3.(2)
Stability ofE2∗: let us consider that one of assumptions (87)-(88) is verified. Then, c(2)<0. If assumption (89) is verified, then, T≠Tl=-(1/λ)ln⁡c(2)/E. Hence, P2(0,T)+Q2(0,T)=E(2)e-λT-c(2)≠0 if assumptions (87)-(88) are verified. Then, let us apply Theorem 4 from [5]. So, let us suppose that 0≤T<Ts(2). It is known that T=0 and E2∗=U2(1) so E2∗ is unstable because ψ(δ)>0 for δ<α2. Taking into account the conclusions of Theorem 4 from [5], one obtains the stability of E2∗.(3)
Let us suppose that the assumptiondet⁡(J)=A11(3)A22(3)-A21(3)A12(3)>0 of (90) is verified. Then, c>B so T≥0>Tl=-1/λln⁡(c/B). Thus, let us apply Theorem 4 from [5]. It is known that when T=0 and E3(3)=U3(1) then E3(3) is stable if assumption (90) is verified and unstable if not. Taking into account the conclusions of Theorem 4 from [5], one gets the stability of E3∗.


## 6.1. Local Stability ofE0∗ and E1∗

We are in situations whereQi(x,T)=0. Let us use the following criterion Routh-Hurwitz: the interior equilibrium Ei∗ is stable if a(i)>0, b(i)>0, c(i)>0, and a(i)b(i)-c(i)>0 and unstable if not.Theorem 9.
Let us suppose thatα2>δ. Then, the equilibria E0∗ and E1∗ of model 1.3 are all unstable.Proof.
Let us consider formula (63)–(67). (1)
The equation ofE0∗ is Δ0(x,T)=x3+a(0)x2+b(0)x+c(0)=0.One has the following:a(0)=λ-(1+α1+α2-δ) and b(0)=(1+α1)(α2-δ)-λ(1+α1+α2-δ). Let us suppose that (a(0)>0 and b(0)>0). Then, one would have (1+α1+α2-δ)<λ<(1+α1)(α2-δ)/(1+α1+α2-δ). Consequently, (1+α1)2+(α2-δ)2+(1+α1)(α2-δ)<0 which is absurd. Hence, the equilibrium E0∗ of model 1.3 is unstable by application of the Routh-Hurwitz criterion.(2)
One hasΔ1(x,T)=x3+a(1)x2+b(1)x+c(1)=0. Meanwhile, α2>δ and, then, c(1)=-λ(1+α1)[β2(1+α1)+α2-δ]<0 so the equilibrium E0∗ of model 1.3 is unstable by application of Routh-Hurwitz criterion.


## 6.2. Local Stability ofE2∗ and E3∗

We are in the situation whereQi(x,T)≠0. In order to assess the influence of delay T over the local stability of model 1.3, we use the results upon the local stability of model obtained taking T=0 into model 1.3 and the results of [5] upon the local stability of the delay systems. Let us consider the characteristic equation (63).(i)
One has the following:Qi(z,T)=0⇒z=-E(i)/D(i)∈R. Then, Pi(x,T) and Qi(x,T) do not have any common imaginary roots.(ii)
One has the following:Pi(x,T) and Qi(x,T) are polynomials with real coefficientssoPi(-iω,T)¯=Pi(iω,T) and Qi(-iω,T)¯=Qi(iω,T).(iii)
If-(1/λ)ln⁡c(i)/E(i)≠T so Pi(0,T)+Qi(0,T)=E(i)e-λT-c(i)≠0.(iv)
Let us posex=ρeiφ. Then, Pi(x,T)/Qi(x,T)~1/ρ4 when ρ→+∞.(68)Thuslimsupx→+∞Qix,TPix,T=0<1.Let us consider the function defined onR by the following: Fi(ω)=Piiω,T2-Qiiω,T2.(69)Then,Fω=biω-ω32+ci-aiω22-e-2λTDiω2+Ei(70)SoFiω=ω6+m2ω4+m1ω2+m0,where(71)m2=ai2-2bi,m1=bi2-2aici-e-2TλDi,m0=ci2-Eie-2λT.Let us determineδ(Ti∗)=sgn⁡(dRe⁡(x)/dT)x=iω(Ti∗) the sign of the real part of a solution x of the characteristic equation Δi(x,T)=0.Lemma 10.
Let us considerx, a solution of the characteristic equation Δi(x,T)=Pi(x,T)+e-xTQi(x,T)=0, with (72)Pix,T=-x3+aix2+bix+ciandQix,T=e-λTDix+Ei.Given ωi=ω(Ti∗) positive root of Fi(ω) and Ti∗ the associated delay certifying the following relation:(73)Ti∗=1ωiarctan⁡aiDi-Eiω3+biEi-ciDiω-Diω4+-aiEi+biDiω2+ciEi+2nπωi,n∈N.Let us pose δ(Ti∗)=sgn⁡(dRe⁡(x)/dT)x=iω(Ti∗). Then,(74)δTi∗=sgn⁡Λ1iH3i+Λ2iH1iH2iwith (75)H1i=λ2+ω2,H2i=λEi-Diω22+Ei+λDi2ω2,H3i=ci-aiω22+bi-ω2ω2,Λ1i=-Ti∗Di2-λ2Di2ω4+λDiEi-Ti∗Ei-λDi2-2λDiEi-λ2Di2ω2+λ3DiEi-λ2Ei2Ti∗,Λ2i=-3aiω4+2ω3+3ci+aibi-2biω2-bici.Proof.
Givenωi=ω(Ti∗) positive root of Fi(ω). Let us assume that there is a delay Ti∗ so that(76)Ti∗=1ωTi∗arctan⁡-Im⁡Piiω,Ti∗/Qiiω,Ti∗Re⁡Piiω,Ti∗/Qiiω,Ti∗+2nπωTi∗,n∈N.(77)Then,Ti∗=1ωiarctan⁡aiDi-Eiω3+biEi-ciDiω-Diω4+-aiEi+biDiω2+ciEi+2nπωi,n∈N.Let us pose(78)δTi∗=sgn⁡dRe⁡xdTx=iωTi∗=sgn⁡dRe⁡xdTx=iωTi∗-1.One has (79)Δix,T=Pix,T+e-xTQix,T=0⟹dPix,TdT-TdxdT+xe-xTQix,T+e-xTdQix,TdT=0.Therefore, Pix,T=-x3+aix2+bix+ci and Qix,T=e-λTDix+Ei. Thus, e-xTDie-λT-TQix,T-3x2+2aix+bi(dx/dT)-e-xTxQi(x,T)+λe-λT(D(i)x+E(i))=0.(80)Hence,dxdT-1=e-xTDie-λT-TQix,T-3x2+2aix+bie-xTxQix,T+λe-λTDix+Ei.Thus,dxdT-1=-Tx+λ+Dix+λDix+Ei-3x2+2aix+bix3+aix2+bix+ciand, meanwhile, δ(Ti∗)=sgn⁡(dRe⁡x/dT)x=iω(Ti∗)-1. Then, δTi∗=sgn⁡Λ1iH3i+Λ2iH1iH2i
withH1i=λ2+ω2, H2i = (λEi-Diω2)2 + (Ei+λDi)2ω2, H3i = (ci-aiω2)2 + (bi-ω)2ω2, Λ1i = -Ti∗Di2-λ2Di2ω4 + λDiEi − Ti∗Ei-λDi2,-2λDiEi − λ2Di2ω2 + λ3DiEi-λ2Ei2Ti∗, and Λ2i = -3a(i)ω4 + 2ω3 + (3c(i)+a(i)b(i)-2b(i))ω2 − b(i)c(i).It is noticed that the analysis of the local stability of the system with delay whenQi(x,T)≠0 depends on the existence of positive root for function Fi(ω). The coefficients m0 and m1, of Fi(ω), depend on T. Therefore, its positive root ω depends on T. Using the Cartan method and Viet formulas, one proves on the one hand that Fi(ω) admits at least a positive root if m0<0. On the other hand, if m0≥0, then, Fi(ω) admits either of two positive roots whereas Fi(ω) does not admit any positive root.The conditionm0<0 is equivalent to the following assumption:(81)0≤T≤Ts=-12λln⁡ci2EiwithEi>0andci2<Ei.Let us denotedet⁡Ji=A11(i)A22(i)-A21(i)A12(i), p=-m1m3+m22/3, and q=-m0m32-2(m2/3)3+m1m2m3/3. Then, one obtains the following proposition.Proposition 11.
Let us suppose thatA13(i)A22(i)>0.
Let us poseλmax(i)=A13(i)A22(i)/det⁡Ji+A13(i)A22(i)2 and Ts(i)=-1/2λln⁡(ci2/E(i)), i=2,3.(1)
Fi(ω) admits at least a positive root if(82)0≤λ<λmaxiand0≤T<Tsi.(2)
Fi(ω) does not admit any positive root if one of the following conditions is verified:  (83)Tsi≤Tand27q2-4p3>0.(84)Tsi≤T,27q2-4p3=0andm1m2-m0>0.(85)Tsi≤T,27q2-4p3<0,m2<0,m1<0andm1m2-m0>0.Now, let us use the results of Theorem4 from [5] to study the stability of these points of equilibriums E2∗ and E3∗.Theorem 12.
One poses(86)x-=β1d2+α1+γd2/d12-β1d2+α1+γd2/d122-4β1d2γd2/d12-12β1d2.Let us consider one of these assumptions (82)–(85) and one of the following conditions:(87)γd2d12<1,(88)γd2d12≥1andmax⁡0;x-≤δα2<1,(89)γd2d12≥1,0≤δα2<min⁡1;x-andT≠-1λln⁡1-d12ψδγα2α2-δ,(90)det⁡J=A113A223-A213A123>0andTrJ=A113+A223<0.With ψ(δ) defined in formula (23), consider the following. (1)
Let us suppose that one of assumptions (83)–(85) is verified for i=2,3. Then, there is no change of stability for equilibrium Ei∗,i=2,3.(2)
Let us suppose that assumptions (82) and one of these assumptions (87)–(89) are verified for i=2.
(a)
Ifδ(T2∗)>0 then the equilibrium E2∗ is unstable for any T∈0;Ts.(b)
Ifδ(T2∗)<0 then the equilibrium E2∗ is unstable if 0≤T<T2∗ and stable if T2∗≤T.(3)
Let us suppose that assumption (82) is verified for i=3.
(a)
In caseδ(T3∗)>0,
(i)
If assumption (90) is verified then the equilibrium E3∗ is stable if 0≤T<T3∗ and unstable if T3∗≤T.(ii)
If assumption (90) is not verified then the equilibrium E3∗ is unstable for any T∈0;Ts.(b)
Consideringδ(T3∗)<0,
(i)
If assumption (90) is verified then the equilibrium E3∗ is stable for every T∈0;Ts.(ii)
If assumption (90) is not verified then the equilibrium E3∗ is unstable if 0≤T<T3∗ and stable if T3∗≤T.
WithTi∗ and δ(Ti∗), i=2,3 are the denotations defined in Lemma 10.Proof.
Let us consider formula (63)–(67). Let us consider the function Fi(ω) defined in relation to the formula (70)-(71) for any point of equilibrium Ei∗, i=2,3. Let us consider assumptions (82). Then, Ts(i) exists so that, for any T≤Ts(i), Fi(ω) admits ωi=ω(Ti∗) a positive root (see Proposition 11). Considering Ti∗ the associated delay determined from Lemma 10, consider the following:(1)
Let us suppose that one of assumptions (83)–(85) is verified in i=2,3. Then, ∀T>Ts(i), Fi(ω) does not admit any positive root. Thus, there is no change of stability of the equilibrium Ei∗,i=2,3.(2)
Stability ofE2∗: let us consider that one of assumptions (87)-(88) is verified. Then, c(2)<0. If assumption (89) is verified, then, T≠Tl=-(1/λ)ln⁡c(2)/E. Hence, P2(0,T)+Q2(0,T)=E(2)e-λT-c(2)≠0 if assumptions (87)-(88) are verified. Then, let us apply Theorem 4 from [5]. So, let us suppose that 0≤T<Ts(2). It is known that T=0 and E2∗=U2(1) so E2∗ is unstable because ψ(δ)>0 for δ<α2. Taking into account the conclusions of Theorem 4 from [5], one obtains the stability of E2∗.(3)
Let us suppose that the assumptiondet⁡(J)=A11(3)A22(3)-A21(3)A12(3)>0 of (90) is verified. Then, c>B so T≥0>Tl=-1/λln⁡(c/B). Thus, let us apply Theorem 4 from [5]. It is known that when T=0 and E3(3)=U3(1) then E3(3) is stable if assumption (90) is verified and unstable if not. Taking into account the conclusions of Theorem 4 from [5], one gets the stability of E3∗.


## 7. Global Stability of Model1.3

Theorem 13.
Let us suppose that model1.3 is permanent and that it admits a unique interior equilibrium point. If(91)δ<α2,β2d2-α2<0,d2<d1,γ<α2,T<ln⁡2λ,12<λ,(92)1+α1d22+α2Mvd22+4d2α121+α1d1+λ+β2δα2+ln⁡2-1<0,(93)β1δα2+γMu+d1<α2Mu+d2,
then the unique interior equilibrium pointE3∗=(u∗;v∗,ku∗) of model 1.3 is globally and asymptotically stable.Proof.
Let us consider model1.3 and E3∗=(u∗;v∗,ku∗) a unique interior equilibrium point. Let us pose U=ln⁡(u/u∗), V=ln⁡v/v∗, and W=ln⁡(w/w∗).
Then,u-u∗=u∗[eU-1], v-v∗=v∗[eV-1], and w-w∗=w∗[eW-1]. Thus, 7.1U˙=u∗-1+β1v2+α1vu∗+d2u+d2+γkvku∗+d1w+d1ueU-1+v∗β1-β1v+v∗+α1u∗+d2-γkku∗+d1eV-1+w∗-d1ku∗+d1ueW-1,V˙=u∗β2-β2v∗u∗+d2+β2u+α2vu∗+d2u+d2eU-1-β2u+α2u∗+d2v∗eV-1,W˙τ=λe-wτkeUτ-1-keWτ-1-e-λTeUτ-T-1.
Let us denoteV1(τ)=Uτ. Consider (94)u,v,w∈A+=u,v,w∈R+3:muε≤u≤Mu,mv≤v≤Mvandmw≤w≤Mw.Then, dV1(τ)/dτ=sgn⁡(U(τ))U˙(τ). Meanwhile, sgn⁡eUτ-1(eUτ-1)=eUτ-1 so the superior derivative of V1(τ) in relation to the time, alongside the solutions of system 7.1, gives (95)D+V1τ≤-1+β1Mv2+α1Mvd22+γkMvmuεd12u-u∗+β1δα2-γkkMu+d1v-v∗+d1kMu+d1Muw-w∗.Let us denote V2(τ)=Vτ. However, (β2u+α2)v/(u∗+d2)(u+d2)≤α2v/(u∗+d2)d2 if β2d2-α2<0. Then,(96)D+V2τ≤β2δα2+α2d22Mvu-u∗-α2Mu+d2v-v∗.Let us note V31(τ)=Wτ. One has muε/u∗≤eU(τ)≤Mu/u∗ and mw/w∗≤eW(τ)≤Mw/w∗. Then, superior derivative of V31(τ) in relation to the time, alongside the solutions of 7.1, results in(97)D+V31τ≤λeUτ-1-λmww∗eWτ-1+Muλe-λTk∫τ-TτeUs-1ds.Let us consider the functional V32(τ)=(Muλe-λT/k)∫τ-Tτ∫vτeUs-1dsdv; then,(98)D+V32τ≤TMuλe-λTkeUτ-1-Muλe-λTk∫τ-TτeUs-1ds.
Let us pose thatV3(τ)=V31(τ)+V32(τ). Thus,(99)D+V3τ≤λ1+TMue-λTkeUτ-1-λeWτ-1.(100)Then,D+V3τ≤λMu1+TMue-λTku-u∗-2λmww-w∗.Let us pose V(τ)=V1(τ)+V2(τ)+V3(τ). Then, from formulas (95), (96), and (100), one obtains(101)D+Vτ≤h1u-u∗+h2v-v∗+h3w-w∗ifβ2d2-α2<0(102)withh1=-1+β1Mv2+α1Mvd22+γkMvmuεd12+β2δα2+α2d22Mv+λMu1+Te-λTkMu,h2=-α2Mu+d2+β1δα2-γkkMu+d1andh3=-2λkmuε+d1kMu+d1Mu.We  have:T<ln⁡2λ⟹e-λTk<1,β1δα2+γMu+d1<α2Mu+d2⟹h2<0,and12<λ⟹h3<0.One has α12/4d2<Mu⇒1/Mu<4d2/α12. Model 1.3 is permanent. So, on the one hand, one has mu(0)≥mu>0 if (β1/d22)Mv2+(α1/d22)Mv<(1+α1)/d22. In addition, mu(0)/2≤muε≤mu(0). Moreover, mu(0)≥2γkMuMv/d1.
Then,γkMv/muεd12≤2γMv/mu(0)d12≤mu(0)/2kMud1<mu(0)/Mud1 if T<(ln⁡2)/λ.
So,D+V(τ)<0 if assumptions (91)–(93) are checked.Corollary 14.
Forr≥14, if the following assumptions are verified:(103)γ≤3+1+2r+r2+r2,(104)d2≥1-8rγ-8r+2r1/2r-4γ-42-8/rγ-3-1+2r+r2+r28,(105)d22r≥α2≥max⁡4d2+44d2-1;4d2+3+1+2r+r2+r24d2-1+γ.(106)2r+r2+r≤α1≤-1+α2-γ4d2-1-3-4d2(107)0<β1<1d2,β2<α2d2,12<λ<1+α1,(108)1-d2Mu+d2<δα2<1,β2<β2max=α2Mud2Mu+d2+δα2-1,(109)d2<d1,γ<α2,T<ln⁡2λ,then, the unique interior equilibrium point E3∗=(u∗;v∗,ku∗) of model 1.3 is globally and asymptotically stable.Proof.
Let us consider the assumptions of Theorem13.(110)Posing:χβ1=d2si0<β1<1d2,d2β1si1d2<β1<4.The model is permanent. Then(111)m0≥mu>0⟺1-χβ1Mu+d2<δα2<1,β2<β2max=α2Muχβ1Mu+d2+δα2-1.
Assumptions (91) and (93) give the following assumption:(112)β1<α2-γMu+d2.Moreover, assumptions (91) and (92) give the following assumption:(113)1+α1d22+2α2d2+4d2α121+α1d1+λ+ln⁡2-1<0.Posing (1+α1)/d22=1/r1, 2α2/d2=1/r2, (4d2/α12)((1+α1)/d1)=1/r3, and 4d2λ/α12=1/r4.
Then, condition (113) becomes (114)1r1+1r2+1r3+1r4<1-ln⁡2.By imposing the condition, r1=r2=r3=r4=r, one obtains r≥14 and assumptions (103)–(109). According to Theorem 13, the unique interior equilibrium point E3∗=(u∗;v∗,ku∗) of model 1.3 is globally and asymptotically stable.


## 8. Numerical Simulations

### 8.1. Limit Cycle

Let us consider the periodp=46 and the following control parameters: (d1,d2,γ,α1,α2,δ,β1,β2)=(1.75;0.25;0.5;3.5;1.75;0.4375;3.617;1.1), λ=0.0000123, and T=45.For the graphic illustration, let us consider the initial conditions(u0,v0,w0)=(6;3;0.00331962170895946), the step h=10-5, and the numbers iterations of time N=4600000. One obtains the maximum values of u, v, and w:(115)umax=66.4527115210953,vmax=2836.21769136928,wmax=0.0367713984036395.The trivial equilibrium points of the model areE0=(0;0.0),E 1 = ( 4.5 ; 0 ; 0.00249006081209868 ), and E2=(0;0.000476921218349683;0).In addition,E∗ = (2.19899664564055; 2.19198952607149; 0.00121680786072133) is the unique interior equilibrium. So, the illustrating Figure 1 is for the initial time t0=0.Figure 1
Trajectories and orbits withu0(θ)=φ(θ)=u0eAθ/(1+u(0)(eAθ-1)), ∀θ[-T;0], where A=(1/2T)ln⁡u0-1/u0 if u(0)≠1 and A=-1/2T if u(0)=1.(i) Interpretation. The trajectories of GDP u and of capital v and w (the effect of delay) are periodic. The orbit starting from the initial condition (u0,v0,w0) gravitates around the equilibrium point E∗ without never reaching it. There is, therefore, a limit cycle around this equilibrium point E∗. The orbit revolves around the equilibrium point. They are moving away from trivial equilibrium points and converge towards the limit cycle around E∗. Indeed, one has λ=0.0000123<λmax(3)=0.2302 and T=45<Ts(3)=305733.267595647. According to Proposition 11, ω=0.000686044679105366 is the positive root of F(ω). One has T3∗=2383.38276771532 and δ(T3∗)<0. So, E∗ is unstable according to Theorem 12 because Tr=3.255>0 for T=0. The trivial equilibrium points of the model are all unstable for T<T3∗<T2∗=4437.41337922542. Hence, we have the graphic illustration.


### 8.2. Global Stability

Let us consider the periodp=25 and the following control parameters: (d1,d2,γ,α1,α2,δ,β1,β2) = (101390.241; 50695.121; 0.5; 1822.498; 1747.022; 1721.154; 9.863.10-6; 7.784.10-6) and λ=0.75 and T=0.462.For the graphic illustration, let us consider the initial conditions(u0,v0,w0)=(5000;8000;1344.89438670147), the step h=10-4, and the numbers iterations of time N=250000. From condition (9) of Theorem 4, the system is permanent if α2>δ,0≤β1<4, and mu(0)≥2γkMuMv/d1. One obtains the maximum and minimum values of u, v, and w for ε=mu(0)-2γkMuMv/d1:(116)umax=1661666.077,vmax=38031.766,wmax=486690.726.umin=muε=1139.233,vmin=750.460,wmin=333.6737.The trivial equilibrium points of the model are E0=(0;0;0),E 1 = ( 1823.498 ; 0 ; 534.090 ), and E2=(0;750.630;0).In addition,E∗=(1796.505;777.6445;526.184) is the unique interior equilibrium point verifying condition (91)–(93) of Theorem 13. Then, E∗ is globally stable. Hence, the illustrating Figure 2 is for the initial time t0=0:Figure 2
Trajectories, orbits, and phase portrait withu0(θ)=φ(θ)=u(0)eAθ/(1+u0eAθ-1), ∀θ[-T;0], where A=1/2Tln⁡u0-1/u0 if u(0)≠1 and A=-1/2T if u(0)=1.(ii) Interpretation. The trajectories of GDP u and capital v and w (the effect of delay) stabilize, respectively, around u∗=1796.505, v∗=777.6445, and w∗=526.184 when t is greater than T. The border dynamics show that when the orbit arrives in a border plan, it remains and converges towards a corresponding equilibrium point (see Orbit.2 to Orbit.4 of Figure 2). Moreover, the trivial equilibrium points of the model are all unstable. So, the orbits, are moving away from the borders plans and converge towards E∗. Hence, E∗ is globally stable, which is illustrated by the figure “phase portrait” of Figure 2.


## 8.1. Limit Cycle

Let us consider the periodp=46 and the following control parameters: (d1,d2,γ,α1,α2,δ,β1,β2)=(1.75;0.25;0.5;3.5;1.75;0.4375;3.617;1.1), λ=0.0000123, and T=45.For the graphic illustration, let us consider the initial conditions(u0,v0,w0)=(6;3;0.00331962170895946), the step h=10-5, and the numbers iterations of time N=4600000. One obtains the maximum values of u, v, and w:(115)umax=66.4527115210953,vmax=2836.21769136928,wmax=0.0367713984036395.The trivial equilibrium points of the model areE0=(0;0.0),E 1 = ( 4.5 ; 0 ; 0.00249006081209868 ), and E2=(0;0.000476921218349683;0).In addition,E∗ = (2.19899664564055; 2.19198952607149; 0.00121680786072133) is the unique interior equilibrium. So, the illustrating Figure 1 is for the initial time t0=0.Figure 1
Trajectories and orbits withu0(θ)=φ(θ)=u0eAθ/(1+u(0)(eAθ-1)), ∀θ[-T;0], where A=(1/2T)ln⁡u0-1/u0 if u(0)≠1 and A=-1/2T if u(0)=1.(i) Interpretation. The trajectories of GDP u and of capital v and w (the effect of delay) are periodic. The orbit starting from the initial condition (u0,v0,w0) gravitates around the equilibrium point E∗ without never reaching it. There is, therefore, a limit cycle around this equilibrium point E∗. The orbit revolves around the equilibrium point. They are moving away from trivial equilibrium points and converge towards the limit cycle around E∗. Indeed, one has λ=0.0000123<λmax(3)=0.2302 and T=45<Ts(3)=305733.267595647. According to Proposition 11, ω=0.000686044679105366 is the positive root of F(ω). One has T3∗=2383.38276771532 and δ(T3∗)<0. So, E∗ is unstable according to Theorem 12 because Tr=3.255>0 for T=0. The trivial equilibrium points of the model are all unstable for T<T3∗<T2∗=4437.41337922542. Hence, we have the graphic illustration.


## 8.2. Global Stability

Let us consider the periodp=25 and the following control parameters: (d1,d2,γ,α1,α2,δ,β1,β2) = (101390.241; 50695.121; 0.5; 1822.498; 1747.022; 1721.154; 9.863.10-6; 7.784.10-6) and λ=0.75 and T=0.462.For the graphic illustration, let us consider the initial conditions(u0,v0,w0)=(5000;8000;1344.89438670147), the step h=10-4, and the numbers iterations of time N=250000. From condition (9) of Theorem 4, the system is permanent if α2>δ,0≤β1<4, and mu(0)≥2γkMuMv/d1. One obtains the maximum and minimum values of u, v, and w for ε=mu(0)-2γkMuMv/d1:(116)umax=1661666.077,vmax=38031.766,wmax=486690.726.umin=muε=1139.233,vmin=750.460,wmin=333.6737.The trivial equilibrium points of the model are E0=(0;0;0),E 1 = ( 1823.498 ; 0 ; 534.090 ), and E2=(0;750.630;0).In addition,E∗=(1796.505;777.6445;526.184) is the unique interior equilibrium point verifying condition (91)–(93) of Theorem 13. Then, E∗ is globally stable. Hence, the illustrating Figure 2 is for the initial time t0=0:Figure 2
Trajectories, orbits, and phase portrait withu0(θ)=φ(θ)=u(0)eAθ/(1+u0eAθ-1), ∀θ[-T;0], where A=1/2Tln⁡u0-1/u0 if u(0)≠1 and A=-1/2T if u(0)=1.(ii) Interpretation. The trajectories of GDP u and capital v and w (the effect of delay) stabilize, respectively, around u∗=1796.505, v∗=777.6445, and w∗=526.184 when t is greater than T. The border dynamics show that when the orbit arrives in a border plan, it remains and converges towards a corresponding equilibrium point (see Orbit.2 to Orbit.4 of Figure 2). Moreover, the trivial equilibrium points of the model are all unstable. So, the orbits, are moving away from the borders plans and converge towards E∗. Hence, E∗ is globally stable, which is illustrated by the figure “phase portrait” of Figure 2.


## 9. Conclusion

The economic basic model of our work was one of dynamics Kaldor at an effective growth rate whose rate of saving is the Holling-II type and investment rate is from Leslie and Gower type modified in dimension two. Taking into account the time required for the saving to ensure its self-financing of all the investments, one obtains a model with a delay of the Kaldor type modified. This model is bounded and admits an attractor unit (set). Thus, under certain conditions, this model with delay is permanent. On the one hand, this permanence appears in the form of stationary growth of the capital stock and the product (stable interior equilibrium point) and, on the other hand, in the form of cyclic growth of the capital and the product (limiting cycle). In other words, this permanence avoids a shortage of the capital stock or the long-term production. In front of some other conditions, the dynamic stability with delay is also global; i.e., it does not depend on the level of the capital stock and the level of the production at the initial time. The consideration of the delay (due to the time for the economy to finance investments) can justify the bifurcation of an economic model of the stationary growth towards cyclic growth (see Theorem12, (3)(a)(i)). However, this delay in the model can stabilize an initially unstable equilibrium of the system (see Theorem 12, (3)(b)(ii)), which represents a major economic interest since it allows the saving to get rid of risks from the cyclic growth.



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*Source: 7212795-2018-08-02.xml*