# Cyclic Growth and Global Stability of Economic Dynamics of Kaldor Type in Two Dimensions

Authors: Aka Fulgence Nindjin; Albin Tetchi N’Guessan; Hypolithe Okou; Kessé Thiban Tia
Journal: International Journal of Differential Equations
 (2017)

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## Abstract

This article proposes nonlinear economic dynamics continuous in two dimensions of Kaldor type, the saving rate and the investment rate, which are functions of ecological origin verifying the nonwasting properties of the resources and economic assumption of Kaldor. The important results of this study contain the notions of bounded solutions, the existence of an attractive set, local and global stability of equilibrium, the system permanence, and the existence of a limit cycle.


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## Body

## 1. Introduction

The nonlinear complex dynamics have been introduced in the analysis of the economic phenomena to explain on the one hand the fluctuations noticed in the study of the chronological series and on the other hand the economic crisis in the capitalist system. The economists Goodwin (1967) and Kaldor (1955-1956) used the dynamic samples to explain that the graphs of cyclic and chaotic evolution are endogeneous to the economic system itself.To simplify things, a great number of these models have been elaborated with more restrictive assumption such as linearity. The challenges of structural reforms of dynamics justify the fact that mathematicians are interested in them. Our contribution will thus consist in proposing economic models inspired by the ecological models whose lessons may be important in terms of analyzing systems and their regulation in this context of climate protection (cf. [1, 2]).The basic economic models that we use in this article are those of Kaldor proposed in the works of Hans-Walter Lorenz (cf. [3]).Our study will consist first of modifying the models of Kaldor conferring them with the ecological properties adapted to economy. Therefore, we propose a dynamic model typical of Kaldor-Holling-2 and Leslie-Gower with some modifications. Next, we will study the qualitative comportment of the model at level 2. We set criteria for which we have on the one hand the marking out of the solutions and the existence of an attractive set and, on the overhand, the local stability of the equilibrium and the permanence of the system.At last we study the global stability of one interior equilibrium through the construction of the Lyapunov function.


## 2. Dynamics of Kaldor Ecologic in Two Dimensions

### 2.1. Economic Dynamics of Type Kaldor with the Effective Growth Rate in Two Dimensions

Let us start by giving the notations and definitions of the rates of parameters and functions in applied economies.(1)
The productivity of capital is the quotient of the GDPY by the capital K. We set that σ=Y/K.(2)
The rate of investment is the quotient of the investmentI by the GDP. We note that f=I/Y.(3)
The saving rate of the GDPϕ is the quotient of the saving S by the GDP. We have ϕ=S/Y.(4)
The rate accumulation of the capital is denoted byh=I/K=σf.(5)
We design, respectively, byϱ and α the monetary depreciation rate and the monetary adjustment coefficient.(6)
The ratio saving-capital will be noted asg=S/K=σϕ.Let us consider the original model of Kaldor (cf. [3] page 44): (1)Y˙=αIY,K-SY,K,K˙=IY,K-ϱK,ϱ,α,Y0,K0∈R+,Y,K∈C10;+∞;R+.In order to establish a connection between the economic models of Kaldor and the ecological patterns, let us give up the coercion of monetary scales by writing investment and saving functions in relation to the investment rate, of capital and saving hoarding. Let us also replace the growth rate of the GDP of model (1) by its effective growth rate (cf. [3, 4]). Therefore we get the following assumption.Assumption 1 (Kaldor with effective growth).
(1) Effective growthY˙/Y=Trend⁡(Y)+α(I-S)/Y with the coefficient of monetary mending, α≥0 and the tendency, Trend⁡(Y), such that Trend⁡(0)≥0.
(2) The ratio saving-capital,g(Y,K), is a function verifying ∂g(Y,K)/∂Y>0.
(3) The investment ratef(Y,K) is a function verifying ∂f(Y,K)/∂Y>0 and there is a threshold Ks≥0 so that ∂f(Y,K)/∂K<0,∀K≥Ks.
(4) The accumulation rate of the capitalh(Y,K) is a function verifying ∂h(Y,K)/∂Y>0, ∂h(Y,K)/∂K<0.With Assumption1, we get below the dynamics of Kaldor with an effective growth rate whose f(Y,K) and g(Y,K) can have ecological properties. (2)Y˙=Trend⁡YY+αfY,KY-αgY,KK,K˙=hY,KK-ϱK,ϱ,α,Y0,K0∈R+,Y,K∈C10;+∞;R+.Interpretation. The dynamics of Kaldor with an effective growth rate (2) present some similarities to the classical ecological dynamics. The function gY(Y,K)=α[f(Y,K)-ϕ(Y,K)]Y define the action of the capital K upon production Y. It stimulates the increase of the production when the investment rate f is superior to the saving rate ϕ and stops it in the end.The functiongK(Y,K)=h(Y,K)K defines in its part the action of the production over the capital. The economic model alone summarizes two types of ecological interaction such as mutualising type and the prey-predators type. Let us consider the economic assumption of ecological inspirations on investment rate f, the rate of capital accumulation h, and the ratio saving-capital g.Assumption 2.
(3)
f
Y
,
K
=
b
~
1
K
1
-
m
2
K
Y
+
C
2
w
i
t
h
b
~
1
K
=
a
1
K
+
b
1
,
(4)
h
Y
,
K
=
b
~
2
Y
1
-
m
2
K
Y
+
C
2
w
i
t
h
b
~
2
Y
=
a
2
Y
+
b
2
,
(5)
g
Y
,
K
=
m
1
Y
Y
+
C
1
,where a1,a2,b1,b2,m1,C1∈R.Obviously, the investments are first funded by capitals. We can thus suppose thatf(Y,K)=b~1(K)>0. Next, the investors adjust the rate in relation to the realities for they hate to invest in vain. However it is not necessary to invest more when net profit is beyond expectancies. Let us estimate the losses of the investment by using the function LG(Y,K)=m2K/Y+C2, where m2 is the maximum value of the losses of the investment rate and C2 is the maximum value of the stock of capital. Then we get f(Y,K)=b~1(K)-LG(Y,K)b~1(K)=b~1(K)(1-m2K/Y+C2). For simplicity, we can take b~1(K)=a1K+b1, where a1 and b1 are the constants depending on the economic policies of investments. Then, we get a rate of investment that verifies the economic requirements (5) of Assumption 1.Concerning the accumulation of capital, it is known thath(Y,K)=σf(Y,K)=b~2(Y)(1-m2K/Y+C2) so b~2(Y)=a1Y+σb1. If σ=(a2-a1)Y+b2/b1 then b~2(Y)=a2Y+b2 with a2 the part of GDP converted in the stock of capital and b2∈R. Therefore h(Y,K) verifies the economic constraint (11) of Assumption 1.Concerning the saving, let us takeS(Y,K)=g(Y)K, where g(Y)=m1Y/Y+C1 is the ratio saving-capital. The function g verifies condition (4) of Assumption 1.Suppose now that the tendency is linear and decreasing:Trend⁡(Y)=a0(1-Y/C0). In fact, the tendency of the growth rate of GDP is at the start, a constant a0 for a given period (forthcoming). But it faces some losses due, for example, to corruption, bribes, slush funds, tax haven, whitening of fraudulent funds, manipulations of accounts and media, and any other harmful activity to the growth of the GDP (cf. [5], pages 11, 15-16, and 35–65). Those losses are estimated to a0Y/C0 with C0 the maximum value (monetary) of the GDP that we can get from this economy for the given period.Then, model (2) becomes the economic dynamics: (6)Y˙=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 indicates the product and K the stock of capital, and Y˙ and K˙ indicate, respectively, the increasing speed of the product and the stock of capital.System (6) defined in this way is a more realistic system. It takes into account a great deal of economic observations of interactions between the product (GDP) and the stock of capital of an economy; namely,(1)
in the absence of the stock of capital, there will not be any explosion of the GDP because the increase of GDP becomes logistic so that, despite the technical progress, the economic production remains limited,(2)
in the absence of the production of the GDP, the economy will not be in short of capital ifb2-ϱ>0 because the evolution is as well logistic due to the diversification of the economy or the opportunities to convert a stock of physical capital in the stock of monetary capital,(3)
when the production (GDP) becomes abundant, there is a “saturation” of the ratio saving-capital and of the investment expressing the adoption of a nonwasting policy of the economic resources,(4)
when the production of the GDP is insufficient, the ratio saving-capital gets adapted and becomes proportional to the available GDP in order to avoid a shortage of production.


### 2.2. Presentation of Model (6) Reduced

In order to facilitate the qualitative study of system (6) that possesses 12 parameters (a0,a1,a2,b1,b2,C0,C1,C2,m1,m2,α,ϱ)∈(R+∗)12, let us change variables by reducing the number of parameters to 8.Definition of New Variables
(7)
τ
=
a
0
t
,
u
τ
=
Y
t
C
0
,
v
τ
=
m
2
C
0
K
t
.Definition of the Parameters
(8)
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
.System (6) becomes then the reduced system: (9)u˙τ=1-uτuτ+β1vτ+α11-vτuτ+d2uτ-γuτvτuτ+d1,v˙τ=β2uτ+α21-vτuτ+d2vτ-δv,u,v∈C1R+,R2u0>0,v0>0,with (d1,d2,γ)∈(R+∗)3 and (β1,β2,α1,α2,δ)∈(R+)5 so that (β2,α2)≠(0,0).


## 2.1. Economic Dynamics of Type Kaldor with the Effective Growth Rate in Two Dimensions

Let us start by giving the notations and definitions of the rates of parameters and functions in applied economies.(1)
The productivity of capital is the quotient of the GDPY by the capital K. We set that σ=Y/K.(2)
The rate of investment is the quotient of the investmentI by the GDP. We note that f=I/Y.(3)
The saving rate of the GDPϕ is the quotient of the saving S by the GDP. We have ϕ=S/Y.(4)
The rate accumulation of the capital is denoted byh=I/K=σf.(5)
We design, respectively, byϱ and α the monetary depreciation rate and the monetary adjustment coefficient.(6)
The ratio saving-capital will be noted asg=S/K=σϕ.Let us consider the original model of Kaldor (cf. [3] page 44): (1)Y˙=αIY,K-SY,K,K˙=IY,K-ϱK,ϱ,α,Y0,K0∈R+,Y,K∈C10;+∞;R+.In order to establish a connection between the economic models of Kaldor and the ecological patterns, let us give up the coercion of monetary scales by writing investment and saving functions in relation to the investment rate, of capital and saving hoarding. Let us also replace the growth rate of the GDP of model (1) by its effective growth rate (cf. [3, 4]). Therefore we get the following assumption.Assumption 1 (Kaldor with effective growth).
(1) Effective growthY˙/Y=Trend⁡(Y)+α(I-S)/Y with the coefficient of monetary mending, α≥0 and the tendency, Trend⁡(Y), such that Trend⁡(0)≥0.
(2) The ratio saving-capital,g(Y,K), is a function verifying ∂g(Y,K)/∂Y>0.
(3) The investment ratef(Y,K) is a function verifying ∂f(Y,K)/∂Y>0 and there is a threshold Ks≥0 so that ∂f(Y,K)/∂K<0,∀K≥Ks.
(4) The accumulation rate of the capitalh(Y,K) is a function verifying ∂h(Y,K)/∂Y>0, ∂h(Y,K)/∂K<0.With Assumption1, we get below the dynamics of Kaldor with an effective growth rate whose f(Y,K) and g(Y,K) can have ecological properties. (2)Y˙=Trend⁡YY+αfY,KY-αgY,KK,K˙=hY,KK-ϱK,ϱ,α,Y0,K0∈R+,Y,K∈C10;+∞;R+.Interpretation. The dynamics of Kaldor with an effective growth rate (2) present some similarities to the classical ecological dynamics. The function gY(Y,K)=α[f(Y,K)-ϕ(Y,K)]Y define the action of the capital K upon production Y. It stimulates the increase of the production when the investment rate f is superior to the saving rate ϕ and stops it in the end.The functiongK(Y,K)=h(Y,K)K defines in its part the action of the production over the capital. The economic model alone summarizes two types of ecological interaction such as mutualising type and the prey-predators type. Let us consider the economic assumption of ecological inspirations on investment rate f, the rate of capital accumulation h, and the ratio saving-capital g.Assumption 2.
(3)
f
Y
,
K
=
b
~
1
K
1
-
m
2
K
Y
+
C
2
w
i
t
h
b
~
1
K
=
a
1
K
+
b
1
,
(4)
h
Y
,
K
=
b
~
2
Y
1
-
m
2
K
Y
+
C
2
w
i
t
h
b
~
2
Y
=
a
2
Y
+
b
2
,
(5)
g
Y
,
K
=
m
1
Y
Y
+
C
1
,where a1,a2,b1,b2,m1,C1∈R.Obviously, the investments are first funded by capitals. We can thus suppose thatf(Y,K)=b~1(K)>0. Next, the investors adjust the rate in relation to the realities for they hate to invest in vain. However it is not necessary to invest more when net profit is beyond expectancies. Let us estimate the losses of the investment by using the function LG(Y,K)=m2K/Y+C2, where m2 is the maximum value of the losses of the investment rate and C2 is the maximum value of the stock of capital. Then we get f(Y,K)=b~1(K)-LG(Y,K)b~1(K)=b~1(K)(1-m2K/Y+C2). For simplicity, we can take b~1(K)=a1K+b1, where a1 and b1 are the constants depending on the economic policies of investments. Then, we get a rate of investment that verifies the economic requirements (5) of Assumption 1.Concerning the accumulation of capital, it is known thath(Y,K)=σf(Y,K)=b~2(Y)(1-m2K/Y+C2) so b~2(Y)=a1Y+σb1. If σ=(a2-a1)Y+b2/b1 then b~2(Y)=a2Y+b2 with a2 the part of GDP converted in the stock of capital and b2∈R. Therefore h(Y,K) verifies the economic constraint (11) of Assumption 1.Concerning the saving, let us takeS(Y,K)=g(Y)K, where g(Y)=m1Y/Y+C1 is the ratio saving-capital. The function g verifies condition (4) of Assumption 1.Suppose now that the tendency is linear and decreasing:Trend⁡(Y)=a0(1-Y/C0). In fact, the tendency of the growth rate of GDP is at the start, a constant a0 for a given period (forthcoming). But it faces some losses due, for example, to corruption, bribes, slush funds, tax haven, whitening of fraudulent funds, manipulations of accounts and media, and any other harmful activity to the growth of the GDP (cf. [5], pages 11, 15-16, and 35–65). Those losses are estimated to a0Y/C0 with C0 the maximum value (monetary) of the GDP that we can get from this economy for the given period.Then, model (2) becomes the economic dynamics: (6)Y˙=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 indicates the product and K the stock of capital, and Y˙ and K˙ indicate, respectively, the increasing speed of the product and the stock of capital.System (6) defined in this way is a more realistic system. It takes into account a great deal of economic observations of interactions between the product (GDP) and the stock of capital of an economy; namely,(1)
in the absence of the stock of capital, there will not be any explosion of the GDP because the increase of GDP becomes logistic so that, despite the technical progress, the economic production remains limited,(2)
in the absence of the production of the GDP, the economy will not be in short of capital ifb2-ϱ>0 because the evolution is as well logistic due to the diversification of the economy or the opportunities to convert a stock of physical capital in the stock of monetary capital,(3)
when the production (GDP) becomes abundant, there is a “saturation” of the ratio saving-capital and of the investment expressing the adoption of a nonwasting policy of the economic resources,(4)
when the production of the GDP is insufficient, the ratio saving-capital gets adapted and becomes proportional to the available GDP in order to avoid a shortage of production.


## 2.2. Presentation of Model (6) Reduced

In order to facilitate the qualitative study of system (6) that possesses 12 parameters (a0,a1,a2,b1,b2,C0,C1,C2,m1,m2,α,ϱ)∈(R+∗)12, let us change variables by reducing the number of parameters to 8.Definition of New Variables
(7)
τ
=
a
0
t
,
u
τ
=
Y
t
C
0
,
v
τ
=
m
2
C
0
K
t
.Definition of the Parameters
(8)
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
.System (6) becomes then the reduced system: (9)u˙τ=1-uτuτ+β1vτ+α11-vτuτ+d2uτ-γuτvτuτ+d1,v˙τ=β2uτ+α21-vτuτ+d2vτ-δv,u,v∈C1R+,R2u0>0,v0>0,with (d1,d2,γ)∈(R+∗)3 and (β1,β2,α1,α2,δ)∈(R+)5 so that (β2,α2)≠(0,0).


## 3. Boundness of Model (9) and Existence of a Positively Invariant Attracting Set

In this section, we give the conditions of the boundness of the capital and the stock of capital justifying the fact that the economic resources are limited.Lemma 3.
The interiorint(R+2) and the boundary ∂(R+2) of the positive quadrant are, respectively, unvarying for system (9).Proof.
Given(u(0),v(0))∈R+2, if 0≤τ<+∞, due to the continuity of u and v over the compact [0;τ] then we have(10)uτ=u0exp⁡∫0τ1-ut+β1vt+α11-vtut+d2-γvtut+d1dt,vτ=v0exp⁡∫0τβ2ut+α21-vtut+d2-δdt.So if(u(0),v(0))=(0,0) then (u(τ),v(τ))=(0;0).Ifu(0)>0 then u(τ)>0 and if v(0)>0 then v(τ)>0, ∀τ∈R+.Thereforeu0,v0∈∂R+2⇒uτ,vτ∈∂R+2, ∀τ∈R+, and (u(0),v(0))∈int(R+2)⇒(u(τ),v(τ))∈int(R+2), ∀τ∈R+.Lemma 4 (cf. [6]).
Given(A,B)∈R+2 and ϕ continuous and derivable function so that there is t0≥0 verifying ϕ(t0)>0, then, ∀t≥t0, (11)dϕdt≤B-Aϕ⟹ϕt≤BA-BA-ϕt0e-At-t0,(12)dϕdt≤B-Aϕ⟹ϕt≤BA1+Aϕt0B-1e-At-t0-1,(13)dϕdt≤ϕB-Aϕ⟹ϕt≤BA1+BAϕt0-1e-Bt-t0-1,(14)dϕdt≥ϕB-Aϕ⟹ϕt≥BA1+BAϕt0-1e-Bt-t0-1.Definition 5 (see [7, 8]).
A solution(u,v)=ϕ(t,t0,u0,v0) of (9) is said to be a boundary in R+2, if there is compact A of R+2 and a time T (T=T(t0,u0,v0)) so that ∀(t0,u0,v0)∈R×R+2, we have (u,v)=ϕ(t,t0,u0,v0)∈A for every t≥t0.Theorem 6.
Let us suppose that0<β1<4. Let us set down(15)Mu=4d2β1+β1d2+2α1β1d2+α12β1d24-β1,L1=4+4-β1Mu2164-β1+Mu+d2β2Mu+1+α2-δ24α2.Let us consider the following set:(16)A=u,v∈R+2,0≤u≤Mu,0≤u+v≤L1.(1)
limsup[u(t)]≤Mu.(2)
limsup[u(t)+v(t)]≤L1.(3)
A is unvarying for model (9).(4)
A is an attractive region for any solution of model (9) from the positive quadrant R+2.Proof.
Let us consider system (9). Let us set down (17)f1u,v=1-uτuτ+β1vτ+α11-vτuτ+d2uτ-γuτvτuτ+d1,f2u,v=β2uτ+α21-vτuτ+d2vτ-δv.(1)
Let us show thatlimsup[u(t)]≤Mu=4d2β1+(β1d2+2α1)β1d2+α12/β1d2(4-β1). We have du/dt=(1-u)u+(β1v+α1)(1-v/u+d2)u-γuv/u+d1and maxv{(β1v+α1)(1-v/u+d2)u}=u[β1(u+d2)+α1]2/4β1(u+d2). Then, du/dt≤B1-A1uu with B1=4d2β1+β1d2+2α1β1d2+α12/4β1d2 and A1=4-β1/4. Then through the application property (13) of Lemma 4, we have ∀t≥0, u(t)≤Mu[1+(B1/A1u(0)-1)e-B1t]-1 with Mu=B1/A1. Therefore, ∀ε>0, ∃T1>0 so that ∀t>T1, u(t)≤Mu+ε.(18)Thenlimsup⁡ut≤Mu=4d2β1+β1d2+2α1β1d2+α12β1d24-β1.So∀t>T1,ut≤Mu.(2)
Let us prove thatlimsup[u(t)+v(t)]≤L1. Let us set down S(t)=u(t)+v(t). We have du/dt≤(B1-A1u)u and dv/dt=[β2u+α2-δ-β2u/u+d2v-α2/u+d2v]v≤[(β2Mu+α2-δ)-α2/Mu+d2v]v; then, we have dS/dt+S(t)≤(1+B1-A1u)u+[(1+β2Mu+α2-δ)-α2/Mu+d2v]v. Now ∀a,b∈R+∗, maxx≥0(b-ax)x=b2/4a. Then dS/dt+S(t)≤L1=(4+(4-β1)Mu)2/16(4-β1)+(Mu+d2)(β2Mu+1+α2-δ)2/4α2. Let us set down A2=1>0 and B2=L1>0 then, dS/dt≤(B2-A2S). Now S(0)>0; then, through the application property (12) of Lemma 4, we have ∀t≥0, S(t)≤L1[1+(S(0)/L1-1)e-t]-1; then ∀ε>0,  ∃T2>0 so that ∀t>T2, S(t)≤L1+ε.(19)SolimsupSt≤L1.Consequently∀t≥T2,ut+vt≤L1.(3)
Let us show thatA is unvarying for model (9). Given (u(0),v(0))∈A, from Lemma 3, of (18) and (19), ∀t≥T=max⁡(T1,T2), u0,v0∈int(A)⇒(u(t),v(t))∈A, (u(0),v(0))∈{0}×]0;L1]⇒(u(t),v(t))∈{0}×]0;L1]⊂A, and (u(0),v(0))∈]0;Mu]×{0}⇒(u(t),v(t))∈]0;Mu]×{0}⊂A. So A is unvarying for model (9).(4)
Let us show thatA is an attractive solution of model (9). Let us show that (u(0),v(0))∈R+2. We deduce from Lemma 3, (18), and (19) that ∀t≥T=max⁡(T1,T2), (u(0),v(0))∈int(R+2)⇒(u(t),v(t))∈A, and (u(0),v(0))∈∂(R+2)⇒(u(t),v(t))∈A, ∀t≥T. Consequently A is an attractive region for any solution of model (9) from the positive quadrant R+2.


## 4. Equilibrium of Model (9)

We are now going to give the conditions of a balanced growth (stationary) of the product and the stock of capital and the quantitative values of the parameters in equilibrium.


### 4.1. Case in Which(β1,β2)≠(0,0)

#### 4.1.1. Trivial Equilibrium

Proposition 7.
(1) If0<α2≤δ then system (9) admits two trivial equilibriums:(20)U0∗=0;0,U1∗=1+α1;0.
(2) Ifα2>δ>0 then system (9) admits three trivial equilibriums:(21)U0∗=0;0,U1∗=1+α1;0,U2∗=0;α2-δd2α2.


#### 4.1.2. Interior Equilibrium

Theorem 8.
LetP(x)=a4x4+a3x3+a2x2+a1x+a0 with (22)a4=-β22,a3=1-d1β22-2β2α2+β2β1δ-γβ2,a2=2β2α21-d1-α22+d1β22+α1δβ2-γβ2α2+β1β2δd1+β2d2+α2-δβ1δ-γβ2,a1=α1δβ2d1+α2+1-d1α22+2β2α2d1+β1δ-γβ2d2α2-δ+β1δd1-γα2β2d2+α2-δ,a0=d1α22+α1α2δd1+β1δd1-γα2d2α2-δ.(1)
System (9) does not admit the interior equilibrium if β2Mu+α2-δ<0.(2)
Any interior equilibriumE∗=(u∗;v∗) of system (9) satisfies the following relations:(23)Pu∗=0,v∗=β2u∗+α2-δβ2u∗+α2u∗+d2withβ2u∗+α2-δ>0.Proof.
Let us consider model (9); then, (β2,α2)≠(0,0). Given E∗=(u∗;v∗) an equilibrium of the model (9),(1)
ifβ2Mu∗+α2-δ<0 then β2u∗+α2-δ<0 so for any u∗≥0, we have v∗<0; therefore the system does not admit any interior equilibrium.(2)
GivenE∗=(u∗;v∗)∈(R+∗)2 an interior equilibrium, then we have(24)1-u∗+β1v∗+α11-v∗u∗+d2-γv∗u∗+d1=0,β2u∗+α21-v∗u∗+d2=δ.Therefore,(25)1-u∗β2u∗+α22u∗+d1+α1δβ2u∗+α2u∗+d1+β1δu∗+d1-γβ2u∗+α2β2u∗+α2-δu∗+d2β2u∗+α22u∗+d1=0,v∗=β2u∗+α2-δβ2u∗+α2u∗+d2.Posing(26)a4=-β22,a3=1-d1β22-2β2α2+β2β1δ-γβ2,a2=2β2α21-d1-α22+d1β22+α1δβ2-γβ2α2+β1β2δd1+β2d2+α2-δβ1δ-γβ2,a1=α1δβ2d1+α2+1-d1α22+2β2α2d1+β1δ-γβ2d2α2-δ+β1δd1-γα2β2d2+α2-δ,a0=d1α22+α1α2δd1+β1δd1-γα2d2α2-δand P(x)=a4x4+a3x3+a2x2+a1x+a0, we obtain(27)E∗=u∗;v∗∈R+∗2⟺Pu∗=0,v∗=β2u∗+α2-δβ2u∗+α2u∗+d2withβ2u∗+α2-δ>0.Corollary 9.
Let us suppose thatα2-δ>0 and considering the polynomial P(x) defined in Theorem 8, let us set down p=a4a2-3a32/8, q=a33/8-a4a3a2/2+a42a1, r=a33/8-a4a3a2/4+a4a32a2/16-3a33/44, A=4r+p2/3, B=-q2-2p3/27+8pr/3, and Δ=27B2-4A3.
System (9) admits a unique interior equilibrium E∗=(u∗,v∗) such that P(u∗)=0 and v∗=(β2u∗+α2-δ)/β2u∗+α2(u∗+d2) in each of the following cases:(a)
Δ>0, a0>0.(b)
Δ=0 and one of the following conditions is verified:
(i)
a0>0, p<0, q>0 and a3<0.(ii)
a0<0, p>0, q>0 and a3>0.(iii)
a0<0, p>0, q<0, a3<0, and a32/4<-(2B/23+2p/3).(iv)
a0>0, p<0, q<0, a3<0 and -(2B/23+2p/3)<a32/4.(v)
a0<0, p>0, q<0, a3>0 and -(2B/23+2p/3)<a32/4.Proof.
GivenP(x)=a4x4+a3x3+a2x2+a1x+a0 the polynomial is defined in Theorem 8 and E∗=(u∗,v∗)∈(R+∗)2 such that P(u∗)=0 and v∗=(β2u∗+α2-δ)/β2u∗+α2(u∗+d2).
Let us posex=1/a4(y-a3/4); then, the equation P(x)=0 will be reduced to (E1):y4+py2+qy+r=0 with p=a4a2-3a32/8, q=a33/8-a4a3a2/2+a42a1, r=a33/8-a4a3a2/4+a4a32a2/16-3a33/44. The characteristic equation of (E1) is (E2):u3-2pu2+(p2-4r)u+q2=0. Let us pose A=4r+p2/3, B=-q2-2p3/27+8pr/3, Δ=27B2-4A3. Given, u1, u2, u3 solutions of (E2) certifying -u1×-u2×-u3=-q, then, y1, y2, y3, and y4, the solutions in C of (E1), are y1=1/2(-u1+-u2+-u3), y2=1/2(-u1--u2--u3), y3=1/2(--u1+-u2--u3), y4=1/2(--u1--u2+-u3).
So, the roots inC of P(x) are xi=1/a4(yi-a3/4) for i∈{1,2,3,4}.
The rootsx1, x2, x3, and x4 in C of P(x) prove the following system: x1x2x3x4=a0/a4 and x1+x2+x3+x4=-a3/a4.
By examining the number of positive roots ofP(x) and knowing that P(u∗)=0 and v∗=(β2u∗+α2-δ)/β2u∗+α2(u∗+d2)>0 because α2-δ>0, we get Corollary 9.


### 4.2. Case in Which(β1,β2)=(0,0)

Considering the conditions(β1,β2)=(0,0) and α1≠0, α2≠0 in system (9), we get the following system: (28)u˙τ=1-uτuτ+α11-vτuτ+d2uτ-γuτvτuτ+d1,v˙τ=α21-vτuτ+d2vτ-δv,u,v∈C1R+,R2,u0>0,v0>0,with (d1,d2,α1,α2,γ)∈R+∗5 and δ∈R+.


#### 4.2.1. Trivial Equilibrium

Proposition 10.
(1) Ifα2≤δ, then, system (28) admits two trivial equilibriums:(29)U0∗=0;0,U1∗=1+α1;0.
(2) Ifα2>δ, then, system (28) admits three trivial equilibriums:(30)U0∗=0;0,U1∗=1+α1;0,U2∗=0;α2-δd2α2.


#### 4.2.2. Interior Equilibrium

Theorem 11.
Given thatP(x)=a2x2+a1x+a0 with (31)a2=-α22,a1=α221-d1+δα1α2-γα2α2-δ,a0=α22d1+δα1α2d1-γα2d2α2-δ(1)
ifα2-δ≤0 then system (28) does not admit any interior equilibriums,(2)
ifα2-δ>0 then any interior equilibrium E∗=(u∗;v∗) of system (28) verifies the following system:(32)Pu∗=0,v∗=α2-δα2u∗+d2.Corollary 12.
Let us suppose thatα2-δ>0 and consider the polynomial P(x) of Theorem 11. System (28) admits a unique interior equilibrium E∗=(u∗,v∗) so that u∗=a1+a12+4α22a0/2α22 and v∗=(α2-δ)/α2(u∗+d2) if one of the following conditions is verified: (i)
δ>α2(γd2-d1)/α1d1+γd2.(ii)
δ<α2(γd2-d1)/α1d1+γd2, a12+4α22a0=0, and a1>0.(iii)
δ=α2(γd2-d1)/α1d1+γd2 and a1>0.Proof.
IfE∗=(u∗,v∗) is an interior equilibrium of (28) and P(x)=-α22x2+a1x+a0, the polynomial, a0 and a1 are stipulated in Theorem 11. Then, P(u∗)=0 and v∗=(α2-δ)/α2(u∗+d2).
We geta0=α22d1+δα1α2d1-γα2d2(α2-δ)=0⇔δ=α2(γd2-d1)/α1d1+γd2.
By examining the number of positive roots ofP(x) in which a0=0 and a0≠0, we obtain researched results.


## 4.1. Case in Which(β1,β2)≠(0,0)

### 4.1.1. Trivial Equilibrium

Proposition 7.
(1) If0<α2≤δ then system (9) admits two trivial equilibriums:(20)U0∗=0;0,U1∗=1+α1;0.
(2) Ifα2>δ>0 then system (9) admits three trivial equilibriums:(21)U0∗=0;0,U1∗=1+α1;0,U2∗=0;α2-δd2α2.


### 4.1.2. Interior Equilibrium

Theorem 8.
LetP(x)=a4x4+a3x3+a2x2+a1x+a0 with (22)a4=-β22,a3=1-d1β22-2β2α2+β2β1δ-γβ2,a2=2β2α21-d1-α22+d1β22+α1δβ2-γβ2α2+β1β2δd1+β2d2+α2-δβ1δ-γβ2,a1=α1δβ2d1+α2+1-d1α22+2β2α2d1+β1δ-γβ2d2α2-δ+β1δd1-γα2β2d2+α2-δ,a0=d1α22+α1α2δd1+β1δd1-γα2d2α2-δ.(1)
System (9) does not admit the interior equilibrium if β2Mu+α2-δ<0.(2)
Any interior equilibriumE∗=(u∗;v∗) of system (9) satisfies the following relations:(23)Pu∗=0,v∗=β2u∗+α2-δβ2u∗+α2u∗+d2withβ2u∗+α2-δ>0.Proof.
Let us consider model (9); then, (β2,α2)≠(0,0). Given E∗=(u∗;v∗) an equilibrium of the model (9),(1)
ifβ2Mu∗+α2-δ<0 then β2u∗+α2-δ<0 so for any u∗≥0, we have v∗<0; therefore the system does not admit any interior equilibrium.(2)
GivenE∗=(u∗;v∗)∈(R+∗)2 an interior equilibrium, then we have(24)1-u∗+β1v∗+α11-v∗u∗+d2-γv∗u∗+d1=0,β2u∗+α21-v∗u∗+d2=δ.Therefore,(25)1-u∗β2u∗+α22u∗+d1+α1δβ2u∗+α2u∗+d1+β1δu∗+d1-γβ2u∗+α2β2u∗+α2-δu∗+d2β2u∗+α22u∗+d1=0,v∗=β2u∗+α2-δβ2u∗+α2u∗+d2.Posing(26)a4=-β22,a3=1-d1β22-2β2α2+β2β1δ-γβ2,a2=2β2α21-d1-α22+d1β22+α1δβ2-γβ2α2+β1β2δd1+β2d2+α2-δβ1δ-γβ2,a1=α1δβ2d1+α2+1-d1α22+2β2α2d1+β1δ-γβ2d2α2-δ+β1δd1-γα2β2d2+α2-δ,a0=d1α22+α1α2δd1+β1δd1-γα2d2α2-δand P(x)=a4x4+a3x3+a2x2+a1x+a0, we obtain(27)E∗=u∗;v∗∈R+∗2⟺Pu∗=0,v∗=β2u∗+α2-δβ2u∗+α2u∗+d2withβ2u∗+α2-δ>0.Corollary 9.
Let us suppose thatα2-δ>0 and considering the polynomial P(x) defined in Theorem 8, let us set down p=a4a2-3a32/8, q=a33/8-a4a3a2/2+a42a1, r=a33/8-a4a3a2/4+a4a32a2/16-3a33/44, A=4r+p2/3, B=-q2-2p3/27+8pr/3, and Δ=27B2-4A3.
System (9) admits a unique interior equilibrium E∗=(u∗,v∗) such that P(u∗)=0 and v∗=(β2u∗+α2-δ)/β2u∗+α2(u∗+d2) in each of the following cases:(a)
Δ>0, a0>0.(b)
Δ=0 and one of the following conditions is verified:
(i)
a0>0, p<0, q>0 and a3<0.(ii)
a0<0, p>0, q>0 and a3>0.(iii)
a0<0, p>0, q<0, a3<0, and a32/4<-(2B/23+2p/3).(iv)
a0>0, p<0, q<0, a3<0 and -(2B/23+2p/3)<a32/4.(v)
a0<0, p>0, q<0, a3>0 and -(2B/23+2p/3)<a32/4.Proof.
GivenP(x)=a4x4+a3x3+a2x2+a1x+a0 the polynomial is defined in Theorem 8 and E∗=(u∗,v∗)∈(R+∗)2 such that P(u∗)=0 and v∗=(β2u∗+α2-δ)/β2u∗+α2(u∗+d2).
Let us posex=1/a4(y-a3/4); then, the equation P(x)=0 will be reduced to (E1):y4+py2+qy+r=0 with p=a4a2-3a32/8, q=a33/8-a4a3a2/2+a42a1, r=a33/8-a4a3a2/4+a4a32a2/16-3a33/44. The characteristic equation of (E1) is (E2):u3-2pu2+(p2-4r)u+q2=0. Let us pose A=4r+p2/3, B=-q2-2p3/27+8pr/3, Δ=27B2-4A3. Given, u1, u2, u3 solutions of (E2) certifying -u1×-u2×-u3=-q, then, y1, y2, y3, and y4, the solutions in C of (E1), are y1=1/2(-u1+-u2+-u3), y2=1/2(-u1--u2--u3), y3=1/2(--u1+-u2--u3), y4=1/2(--u1--u2+-u3).
So, the roots inC of P(x) are xi=1/a4(yi-a3/4) for i∈{1,2,3,4}.
The rootsx1, x2, x3, and x4 in C of P(x) prove the following system: x1x2x3x4=a0/a4 and x1+x2+x3+x4=-a3/a4.
By examining the number of positive roots ofP(x) and knowing that P(u∗)=0 and v∗=(β2u∗+α2-δ)/β2u∗+α2(u∗+d2)>0 because α2-δ>0, we get Corollary 9.


## 4.1.1. Trivial Equilibrium

Proposition 7.
(1) If0<α2≤δ then system (9) admits two trivial equilibriums:(20)U0∗=0;0,U1∗=1+α1;0.
(2) Ifα2>δ>0 then system (9) admits three trivial equilibriums:(21)U0∗=0;0,U1∗=1+α1;0,U2∗=0;α2-δd2α2.


## 4.1.2. Interior Equilibrium

Theorem 8.
LetP(x)=a4x4+a3x3+a2x2+a1x+a0 with (22)a4=-β22,a3=1-d1β22-2β2α2+β2β1δ-γβ2,a2=2β2α21-d1-α22+d1β22+α1δβ2-γβ2α2+β1β2δd1+β2d2+α2-δβ1δ-γβ2,a1=α1δβ2d1+α2+1-d1α22+2β2α2d1+β1δ-γβ2d2α2-δ+β1δd1-γα2β2d2+α2-δ,a0=d1α22+α1α2δd1+β1δd1-γα2d2α2-δ.(1)
System (9) does not admit the interior equilibrium if β2Mu+α2-δ<0.(2)
Any interior equilibriumE∗=(u∗;v∗) of system (9) satisfies the following relations:(23)Pu∗=0,v∗=β2u∗+α2-δβ2u∗+α2u∗+d2withβ2u∗+α2-δ>0.Proof.
Let us consider model (9); then, (β2,α2)≠(0,0). Given E∗=(u∗;v∗) an equilibrium of the model (9),(1)
ifβ2Mu∗+α2-δ<0 then β2u∗+α2-δ<0 so for any u∗≥0, we have v∗<0; therefore the system does not admit any interior equilibrium.(2)
GivenE∗=(u∗;v∗)∈(R+∗)2 an interior equilibrium, then we have(24)1-u∗+β1v∗+α11-v∗u∗+d2-γv∗u∗+d1=0,β2u∗+α21-v∗u∗+d2=δ.Therefore,(25)1-u∗β2u∗+α22u∗+d1+α1δβ2u∗+α2u∗+d1+β1δu∗+d1-γβ2u∗+α2β2u∗+α2-δu∗+d2β2u∗+α22u∗+d1=0,v∗=β2u∗+α2-δβ2u∗+α2u∗+d2.Posing(26)a4=-β22,a3=1-d1β22-2β2α2+β2β1δ-γβ2,a2=2β2α21-d1-α22+d1β22+α1δβ2-γβ2α2+β1β2δd1+β2d2+α2-δβ1δ-γβ2,a1=α1δβ2d1+α2+1-d1α22+2β2α2d1+β1δ-γβ2d2α2-δ+β1δd1-γα2β2d2+α2-δ,a0=d1α22+α1α2δd1+β1δd1-γα2d2α2-δand P(x)=a4x4+a3x3+a2x2+a1x+a0, we obtain(27)E∗=u∗;v∗∈R+∗2⟺Pu∗=0,v∗=β2u∗+α2-δβ2u∗+α2u∗+d2withβ2u∗+α2-δ>0.Corollary 9.
Let us suppose thatα2-δ>0 and considering the polynomial P(x) defined in Theorem 8, let us set down p=a4a2-3a32/8, q=a33/8-a4a3a2/2+a42a1, r=a33/8-a4a3a2/4+a4a32a2/16-3a33/44, A=4r+p2/3, B=-q2-2p3/27+8pr/3, and Δ=27B2-4A3.
System (9) admits a unique interior equilibrium E∗=(u∗,v∗) such that P(u∗)=0 and v∗=(β2u∗+α2-δ)/β2u∗+α2(u∗+d2) in each of the following cases:(a)
Δ>0, a0>0.(b)
Δ=0 and one of the following conditions is verified:
(i)
a0>0, p<0, q>0 and a3<0.(ii)
a0<0, p>0, q>0 and a3>0.(iii)
a0<0, p>0, q<0, a3<0, and a32/4<-(2B/23+2p/3).(iv)
a0>0, p<0, q<0, a3<0 and -(2B/23+2p/3)<a32/4.(v)
a0<0, p>0, q<0, a3>0 and -(2B/23+2p/3)<a32/4.Proof.
GivenP(x)=a4x4+a3x3+a2x2+a1x+a0 the polynomial is defined in Theorem 8 and E∗=(u∗,v∗)∈(R+∗)2 such that P(u∗)=0 and v∗=(β2u∗+α2-δ)/β2u∗+α2(u∗+d2).
Let us posex=1/a4(y-a3/4); then, the equation P(x)=0 will be reduced to (E1):y4+py2+qy+r=0 with p=a4a2-3a32/8, q=a33/8-a4a3a2/2+a42a1, r=a33/8-a4a3a2/4+a4a32a2/16-3a33/44. The characteristic equation of (E1) is (E2):u3-2pu2+(p2-4r)u+q2=0. Let us pose A=4r+p2/3, B=-q2-2p3/27+8pr/3, Δ=27B2-4A3. Given, u1, u2, u3 solutions of (E2) certifying -u1×-u2×-u3=-q, then, y1, y2, y3, and y4, the solutions in C of (E1), are y1=1/2(-u1+-u2+-u3), y2=1/2(-u1--u2--u3), y3=1/2(--u1+-u2--u3), y4=1/2(--u1--u2+-u3).
So, the roots inC of P(x) are xi=1/a4(yi-a3/4) for i∈{1,2,3,4}.
The rootsx1, x2, x3, and x4 in C of P(x) prove the following system: x1x2x3x4=a0/a4 and x1+x2+x3+x4=-a3/a4.
By examining the number of positive roots ofP(x) and knowing that P(u∗)=0 and v∗=(β2u∗+α2-δ)/β2u∗+α2(u∗+d2)>0 because α2-δ>0, we get Corollary 9.


## 4.2. Case in Which(β1,β2)=(0,0)

Considering the conditions(β1,β2)=(0,0) and α1≠0, α2≠0 in system (9), we get the following system: (28)u˙τ=1-uτuτ+α11-vτuτ+d2uτ-γuτvτuτ+d1,v˙τ=α21-vτuτ+d2vτ-δv,u,v∈C1R+,R2,u0>0,v0>0,with (d1,d2,α1,α2,γ)∈R+∗5 and δ∈R+.


### 4.2.1. Trivial Equilibrium

Proposition 10.
(1) Ifα2≤δ, then, system (28) admits two trivial equilibriums:(29)U0∗=0;0,U1∗=1+α1;0.
(2) Ifα2>δ, then, system (28) admits three trivial equilibriums:(30)U0∗=0;0,U1∗=1+α1;0,U2∗=0;α2-δd2α2.


### 4.2.2. Interior Equilibrium

Theorem 11.
Given thatP(x)=a2x2+a1x+a0 with (31)a2=-α22,a1=α221-d1+δα1α2-γα2α2-δ,a0=α22d1+δα1α2d1-γα2d2α2-δ(1)
ifα2-δ≤0 then system (28) does not admit any interior equilibriums,(2)
ifα2-δ>0 then any interior equilibrium E∗=(u∗;v∗) of system (28) verifies the following system:(32)Pu∗=0,v∗=α2-δα2u∗+d2.Corollary 12.
Let us suppose thatα2-δ>0 and consider the polynomial P(x) of Theorem 11. System (28) admits a unique interior equilibrium E∗=(u∗,v∗) so that u∗=a1+a12+4α22a0/2α22 and v∗=(α2-δ)/α2(u∗+d2) if one of the following conditions is verified: (i)
δ>α2(γd2-d1)/α1d1+γd2.(ii)
δ<α2(γd2-d1)/α1d1+γd2, a12+4α22a0=0, and a1>0.(iii)
δ=α2(γd2-d1)/α1d1+γd2 and a1>0.Proof.
IfE∗=(u∗,v∗) is an interior equilibrium of (28) and P(x)=-α22x2+a1x+a0, the polynomial, a0 and a1 are stipulated in Theorem 11. Then, P(u∗)=0 and v∗=(α2-δ)/α2(u∗+d2).
We geta0=α22d1+δα1α2d1-γα2d2(α2-δ)=0⇔δ=α2(γd2-d1)/α1d1+γd2.
By examining the number of positive roots ofP(x) in which a0=0 and a0≠0, we obtain researched results.


## 4.2.1. Trivial Equilibrium

Proposition 10.
(1) Ifα2≤δ, then, system (28) admits two trivial equilibriums:(29)U0∗=0;0,U1∗=1+α1;0.
(2) Ifα2>δ, then, system (28) admits three trivial equilibriums:(30)U0∗=0;0,U1∗=1+α1;0,U2∗=0;α2-δd2α2.


## 4.2.2. Interior Equilibrium

Theorem 11.
Given thatP(x)=a2x2+a1x+a0 with (31)a2=-α22,a1=α221-d1+δα1α2-γα2α2-δ,a0=α22d1+δα1α2d1-γα2d2α2-δ(1)
ifα2-δ≤0 then system (28) does not admit any interior equilibriums,(2)
ifα2-δ>0 then any interior equilibrium E∗=(u∗;v∗) of system (28) verifies the following system:(32)Pu∗=0,v∗=α2-δα2u∗+d2.Corollary 12.
Let us suppose thatα2-δ>0 and consider the polynomial P(x) of Theorem 11. System (28) admits a unique interior equilibrium E∗=(u∗,v∗) so that u∗=a1+a12+4α22a0/2α22 and v∗=(α2-δ)/α2(u∗+d2) if one of the following conditions is verified: (i)
δ>α2(γd2-d1)/α1d1+γd2.(ii)
δ<α2(γd2-d1)/α1d1+γd2, a12+4α22a0=0, and a1>0.(iii)
δ=α2(γd2-d1)/α1d1+γd2 and a1>0.Proof.
IfE∗=(u∗,v∗) is an interior equilibrium of (28) and P(x)=-α22x2+a1x+a0, the polynomial, a0 and a1 are stipulated in Theorem 11. Then, P(u∗)=0 and v∗=(α2-δ)/α2(u∗+d2).
We geta0=α22d1+δα1α2d1-γα2d2(α2-δ)=0⇔δ=α2(γd2-d1)/α1d1+γd2.
By examining the number of positive roots ofP(x) in which a0=0 and a0≠0, we obtain researched results.


## 5. Local Stability and Permanence of Model (9)

In this section, we first define the conditions in which this balanced growth of the product and the stock of capital of the economy are stable or unstable. Then, let us examine the possibility of having permanently those two parameters of the economy (sustainable development). This permanence of the product and the stock of capital of the economy are noticed either through the convergence (of both parameters) towards a stable equilibrium or through a fluctuation of both parameters around an unstable equilibrium, that is, the convergence towards a limited cycle.


### 5.1. Local Stability of Model (9)

In system (9), we pose the following: (33)f1u,v=1-uτuτ+β1vτ+α11-vτuτ+d2uτ-γuτvτuτ+d1,f2u,v=β2uτ+α21-vτuτ+d2vτ-δv.We have u˙(τ)=f1(u,v) and v˙(τ)=f2(u,v).Let us note thatJ(E), the Jacobian matrix of the system, is linear around E=(u,v). Then, we have (34)JE=∂f1∂uE∂f1∂vE∂f2∂uE∂f2∂vE=J11EJ12EJ21EJ22Ewith(35)∂f1∂u=1-2u+β1v+α11-vu+d2+β1v+α1uvu+d22-γd1vu+d12,∂f1∂v=β1-2β1u+d2v-α1u+d2-γu+d1u,∂f2∂u=β21-vu+d2+β2u+α2vu+d22v,∂f2∂v=β2u+α21-2vu+d2-δ.Theorem 13 (local stability).
(1) Stability of U0∗=(0;0):(a)
U0∗ is an unstable node if α2>δ.(b)
U0∗ is an unstable saddle point if α2<δ,
(i)
repulsive along the directionu,(ii)
attractive along the directionv.
(2) Stability of U1∗=(1+α1;0): (a)
U1∗ is stable if α2+(1+α1)β2<δ.(b)
U1∗ is an unstable saddle point if α2+(1+α1)β2>δ,
(i)
attractive along the directionu,(ii)
repulsive along the directionw1.
(3) Stability of U2∗=(0;(α2-δ)d2/α2) for α2>δ: (a)
U2∗ is stable if Q(δ)<0.(b)
U2∗ is an unstable saddle point if Q(δ)>0, repulsive along the direction w2 and attractive along the direction v.
(4) Given E∗=(u∗;v∗), an interior equilibrium of (9) verifying system (23) of Theorem 8, and J its associate Jacobian matrix, (a)
E∗ is stable (node or a focus) if det(J)>0 and Tr(J)<0,(b)
E∗ is marginal or a center if det(J)≥0 and Tr(J)=0,(c)
E∗ is unstable if det(J)<0 or det(J)>0 and Tr(J)>0; more precisely,
(i)
E∗ is a node or a focus if det(J)>0 and Tr(J)>0,(ii)
E∗ is a unstable saddle if det(J)<0.With Tr(J) the trace and det(J) the determinant of J, the polynomial Q(x)=-d2β1d1x2+[γα2d2+d2d1β1α2+d1α1α2]x+α22[d1-γd2], the vectors w1=(1+α1)[β1-α1/1+α1+d2-γ/1+α1+d1];(1+α1)(β2+1)+α2-δ, and w2=Q(δ)/d1α22+α2-δ;[β2d2δ+α2(α2-δ)](α2-δ)/α22.Proof.
GivenJk, the Jacobian matrix of the system, is linear around the equilibrium Uk∗ for k∈{0,1,2} and J of E∗=(u∗,v∗), we have the following:(1)
Stability ofU0∗=(0;0):We have(36)U0∗=0;0⟹J0=1+α100α2-δ.The numbersλ1=1+α1>0 and λ2=α2-δ are the eigenvalues of J0. Eigenspace associated with λ1 is Eλ1(0)=〈(1;0)〉, where 〈{rk}1≤k≤N〉 indicates the vectorial subspace generated by the family: {rk}1≤k≤N∗ with N∈N. The eigenspace associated with λ2 is Eλ2(0)=〈(0;1)〉. We have the following:
(a)
Ifα2>δ then λ1>0 and λ2>0; then U0∗ is an unstable node and its unstable manifold is Eu=〈(1;0),(0;1)〉=R2.(b)
Ifα2<δ then λ1>0 and λ2<0; then U0∗ is an unstable saddle point, the unstable manifold of which is Eu=〈(1;0)〉 and the stable manifold is Es=〈(0;1)〉.(c)
Ifα2=δ then λ1>0 and λ2=0; therefore U0∗ is an equilibrium the unstable manifold of which is Eu=〈(1;0)〉 and of central manifold is Ec=〈(0;1)〉.(2)
Stability ofU1∗=(1+α1;0):We have(37)J1=-1+α11+α1β1-α11+α1+d2-γ1+α1+d101+α1β2+α2-δ.Soλ1=-(1+α1)<0 and λ2=(1+α1)β2+α2-δ are the eigenvalues of J1.The eigenspace associated withλ1 is Eλ1(1)=〈(1;0)〉.Note thatw1=(1+α1)[β1-α1/1+α1+d2-γ/1+α1+d1];(1+α1)(β2+1)+α2-δ.The eigenspace associated withλ2 is Eλ21=w1.We get the following:
(a)
Ifα2+(1+α1)β2<δ then λ1<0 and λ2<0; then U1∗ is stable and its stable manifold is Eu=R2.(b)
Ifα2+(1+α1)β2>δ then λ1<0 and λ2>0; then U1∗ is an unstable saddle point of which the stable manifold is Es=〈(1;0)〉 and the unstable manifold is Eu=〈w1〉.(c)
Ifα2+(1+α1)β2=δ then λ1<0 and λ2=0; consequently U1∗ is an equilibrium the stable manifold of which is Es=〈(1;0)〉 and the central manifold is Ec=〈w1〉.(3)
Stability ofU2∗=0;α2-δd2/α2:We have(38)J2=Qδd1α220β2d2δ+α2α2-δα2-δα22-α2-δ.Soλ1=Q(δ)/d1α22 and λ2=-(α2-δ) are the eigenvalues of J2.Let us note thatw2=Q(δ)/d1α22+α2-δ;[β2d2δ+α2(α2-δ)](α2-δ)/α22.The eigenspace associated withλ1 is Eλ1(2)=〈w2〉.The eigenspace associated withλ2 is Eλ2(2)=〈(0,1)〉. We get the following:
(a)
IfQ(δ)<0 then λ1<0 and λ2<0; then, U2∗ is stable and its stable manifold is Es=R2.(b)
IfQ(δ)≥0 then λ1<0 and λ2≥0; then U2∗ is an unstable saddle the unstable manifold of which is Eu=〈w2〉 and the stable manifold of which is Es=〈(0,1)〉.(4)
Stability ofE∗=(u∗;v∗):Letλ1∗ and λ2∗ be the eigenvalues of J. Thus, let us note Tr(J)=λ1∗+λ2∗ its trace and det(J)=λ1∗λ2∗ its determinant. The eigenspace associated with λ1∗ is Eλ1∗=〈J12;λ1∗-J11〉 and the eigenspace associated with λ2∗ is Eλ2∗=〈J21;λ2∗-J22〉.
(a)
Ifdet(J)>0 and Tr(J)<0 then Re[λ1∗]<0 and Re[λ2∗]<0. So, E∗ is stable (stable node or stable focus).(b)
Ifdet(J)≥0 and Tr(J)=0 then λ1∗,λ2∗∈iR. Therefore E∗ is a marginal or a center.(c)
Ifdet(J)<0 or det(J)>0 and Tr(J)>0 then E∗ is unstable. In fact,
(i)
ifdet(J)>0 and Tr(J)>0 then Re[λ1∗]>0 and Re[λ2∗]>0 or λ1∗ and λ2∗ are conjugated complexes. So, E∗ is node or an unstable center,(ii)
ifdet(J)<0 then E∗ is an unstable saddle.


### 5.2. Permanence of Model (9)

Definition 14 (see [8]).
Given solutionu=(x1,x2,…,xn) of a differential system(39)u˙=ft,u,(1)
a componentxi of the solution u of (39) is said to beweakly persistent if limsup[xi(t)]>0,(2)
a componentxi of the solution u of (39) is said to behighly persistent if liminf[xi(t)]>0,(3)
a componentxi of the solution u of (39) is said to beuniformly persistent if there is ε such that liminf[xi(t)]≥ε>0,(4)
System (39) is said to bedissipative as for any component xi of the solution there is a constant Mi>0 such that limsup[xi(t)]≥Mi>0,(5)
System (39) is said to bepermanent if it isuniformly persistent anddissipative.LetΩ¯ be complete metric space and Ω for an open set such that Ω¯=Ω∪∂Ω. Further, we shall take Ω=int(R+2).Definition 15 (see [8]).
A flow or semiflow onΩ¯ under which Ω and ∂Ω are forward invariant is said to be permanent if it is dissipative and if there is a number ε>0 such that any trajectory starting in Ω will be at least at a distance ε from ∂Ω for all sufficiently large t.Definition 16 (see [8]).
(1) Theω-limit set ω(∂Ω) is said to be isolated if it has a covering M=⋃k=1NMk of pairwise disjoint sets Mk which are isolated and invariant with respect to the flow or the semiflow both on ∂Ω and on Ω¯=Ω∪∂Ω.
(2) The setω(∂Ω) is said to be acyclic if there exists an isolated covering M=⋃k=1NMk such that no subset of Mk is a cycle.Lemma 17 (see [8]).
Suppose that a semiflow onΩ¯ leaves both Ω and ∂Ω forward invariant, maps bounded sets in Ω¯ to precompact set for t>0, and it is dissipative. If in addition (1)
ω(∂Ω) is isolated and acyclic,(2)
Es[Mk]∩Ω=∅ for all k∈{1;2;…;N}, where M=⋃k=1NMk is the isolated covering used in the definition of acyclicity of ω(∂Ω) and Es denotes the stable manifold,then the semiflow is permanent.Theorem 18.
Let us assume that0<β1<4 and α2>δ; then we pose (40)Mu=4d2β1+β1d2+2α1β1d2+α12β1d24-β1;L1=4+4-β1Mu2164-β1+Mu+d2β2Mu+1+α2-δ24α2;mu=-β1d2L12-α1d2+γd1L1+1+α1;mv=α2-δd2β2d2+α2;Qx=-d2β1d1x2+γα2d2+d2d1β1α2+d1α1α2x+α22d1-γd2.Let us consider the following assumptions: (41)H1-β1d2L12-α1d2+γd1L1+1+α1>0;H2Qδ>0.Under the assumptions (H1) and (H2), model (9) is permanent and any positive solution (u,v) of (9) verifies(42)mu≤u≤Mu;mv≤v≤L1-mu.Proof.
GivenΩ=int⁡(R+2) and ∂Ω=〈(1;0)〉∪〈(0;1)〉, its frontier, and Ω¯=Ω∪∂Ω, we know that Ω and ∂Ω are invariants for model (9) (cf. Lemma 3) and that A is attractive bounded for any trajectory from Ω¯=R+2 (cf. Theorem 6). Let us assume that 0≤β1<4 and α2>δ and let us apply Lemma 17: (1)
Let us justify the fact that model (9) is dissipative over Ω¯=R+2:
(a)
Let us show thatliminf[u(t)]≥mu=-β1/d2L12-(α1/d2+γ/d1)L1+1+α1. Posing A1=1 and B1=1+α1-L1(β1L1+α1)/d2-γL1/d1, we have du/dt≥(B1-A1u)u and u(0)>0. So, from property (14) of Lemma 4, we have liminf[u(t)]≥mu=-β1/d2L12-(α1/d2+γ/d1)L1+1+α1.(b)
Let us show thatliminf[v(t)]≥mv=(α2-δ)d2/β2d2+α2. Posing B3=(α2-δ) and A3=(β2+α2/d2), then, we have dv/dt≥(B3-A3v)v and v(0)>0. We get liminf[v(t)]≥mv=(α2-δ)d2/β2d2+α2>0 for α2>δ.(c)
We deduce that model (9) is dissipative over Ω¯=R+2 as soon as the assumption (H1) is verified.(2)
Let us prove thatω(∂Ω) is isolated and acyclic. We have ∂Ω=〈(1;0)〉∪〈(0;1)〉. On the one hand, ω[〈(1;0)〉]={U0∗;U1∗}⊂〈(1;0)〉; now the stable manifold U1∗ is Es[U1∗]=〈(1,0)〉 if α2+(1+α1)β2>δ and U0∗ is unstable if α2≥δ. Then any trajectory from 〈(1;0)〉 other than U0∗ approaches U1∗ if α2≥δ for α2≥δ⇒α2+(1+α1)β2>δ. On the other hand, ω[〈(0;1)〉]={U0;U2}⊂〈(0;1)〉; now the stable manifold of U2∗ is Es[U2∗]=〈(0,1)〉 if Q(δ)>0 and U0∗ is unstable if α2≥δ; then any trajectory from 〈(0;1)〉 other than U0∗ approaches U2∗ if α2≥δ and Q(δ)>0. Given that α2>δ then we deduce that ω[∂Ω]={U0;U1;U2}⊂∂Ω is isolated and acyclic if Q(δ)>0 (if the assumption (H2) is verified).(3)
Let us justify thatEs[ω(∂Ω)]∩Ω=∅. We have ω[∂Ω]={U0;U1;U2} and Ω=int⁡(R+2). If α2≥δ, α2+(1+α1)β2>δ, and Q(δ)>0 then Es[ω(∂Ω)]∩Ω=∅; now α2>δ. So Es[ω(∂Ω)]∩Ω=∅ if Q(δ)>0.Definitively, system (9) is permanent.


### 5.3. Limit Cycle of Model (9)

The theorem below presents the conditions for a cyclic growth of the product and the stock of capital of the economy.Theorem 19.
Let us recall the notations of Theorem13. Let us suppose that the assumptions of Theorem 18 are verified and that model (9) admits a unique interior equilibrium E∗=(u∗;v∗). If det⁡(J)>0 and Tr⁡(J)>0 then model (9) admits a limit cycle contained in the attractive region A.Proof.
Under the assumption of Theorem18, model (9) is permanent and if det⁡(J)>0 and Tr⁡(J)>0 then the unique interior equilibrium E∗=(u∗;v∗) is unstable so model (9) admits a limit cycle contained in the compact and bounded region A (from Poincaré-Beddicton’s theorem).


## 5.1. Local Stability of Model (9)

In system (9), we pose the following: (33)f1u,v=1-uτuτ+β1vτ+α11-vτuτ+d2uτ-γuτvτuτ+d1,f2u,v=β2uτ+α21-vτuτ+d2vτ-δv.We have u˙(τ)=f1(u,v) and v˙(τ)=f2(u,v).Let us note thatJ(E), the Jacobian matrix of the system, is linear around E=(u,v). Then, we have (34)JE=∂f1∂uE∂f1∂vE∂f2∂uE∂f2∂vE=J11EJ12EJ21EJ22Ewith(35)∂f1∂u=1-2u+β1v+α11-vu+d2+β1v+α1uvu+d22-γd1vu+d12,∂f1∂v=β1-2β1u+d2v-α1u+d2-γu+d1u,∂f2∂u=β21-vu+d2+β2u+α2vu+d22v,∂f2∂v=β2u+α21-2vu+d2-δ.Theorem 13 (local stability).
(1) Stability of U0∗=(0;0):(a)
U0∗ is an unstable node if α2>δ.(b)
U0∗ is an unstable saddle point if α2<δ,
(i)
repulsive along the directionu,(ii)
attractive along the directionv.
(2) Stability of U1∗=(1+α1;0): (a)
U1∗ is stable if α2+(1+α1)β2<δ.(b)
U1∗ is an unstable saddle point if α2+(1+α1)β2>δ,
(i)
attractive along the directionu,(ii)
repulsive along the directionw1.
(3) Stability of U2∗=(0;(α2-δ)d2/α2) for α2>δ: (a)
U2∗ is stable if Q(δ)<0.(b)
U2∗ is an unstable saddle point if Q(δ)>0, repulsive along the direction w2 and attractive along the direction v.
(4) Given E∗=(u∗;v∗), an interior equilibrium of (9) verifying system (23) of Theorem 8, and J its associate Jacobian matrix, (a)
E∗ is stable (node or a focus) if det(J)>0 and Tr(J)<0,(b)
E∗ is marginal or a center if det(J)≥0 and Tr(J)=0,(c)
E∗ is unstable if det(J)<0 or det(J)>0 and Tr(J)>0; more precisely,
(i)
E∗ is a node or a focus if det(J)>0 and Tr(J)>0,(ii)
E∗ is a unstable saddle if det(J)<0.With Tr(J) the trace and det(J) the determinant of J, the polynomial Q(x)=-d2β1d1x2+[γα2d2+d2d1β1α2+d1α1α2]x+α22[d1-γd2], the vectors w1=(1+α1)[β1-α1/1+α1+d2-γ/1+α1+d1];(1+α1)(β2+1)+α2-δ, and w2=Q(δ)/d1α22+α2-δ;[β2d2δ+α2(α2-δ)](α2-δ)/α22.Proof.
GivenJk, the Jacobian matrix of the system, is linear around the equilibrium Uk∗ for k∈{0,1,2} and J of E∗=(u∗,v∗), we have the following:(1)
Stability ofU0∗=(0;0):We have(36)U0∗=0;0⟹J0=1+α100α2-δ.The numbersλ1=1+α1>0 and λ2=α2-δ are the eigenvalues of J0. Eigenspace associated with λ1 is Eλ1(0)=〈(1;0)〉, where 〈{rk}1≤k≤N〉 indicates the vectorial subspace generated by the family: {rk}1≤k≤N∗ with N∈N. The eigenspace associated with λ2 is Eλ2(0)=〈(0;1)〉. We have the following:
(a)
Ifα2>δ then λ1>0 and λ2>0; then U0∗ is an unstable node and its unstable manifold is Eu=〈(1;0),(0;1)〉=R2.(b)
Ifα2<δ then λ1>0 and λ2<0; then U0∗ is an unstable saddle point, the unstable manifold of which is Eu=〈(1;0)〉 and the stable manifold is Es=〈(0;1)〉.(c)
Ifα2=δ then λ1>0 and λ2=0; therefore U0∗ is an equilibrium the unstable manifold of which is Eu=〈(1;0)〉 and of central manifold is Ec=〈(0;1)〉.(2)
Stability ofU1∗=(1+α1;0):We have(37)J1=-1+α11+α1β1-α11+α1+d2-γ1+α1+d101+α1β2+α2-δ.Soλ1=-(1+α1)<0 and λ2=(1+α1)β2+α2-δ are the eigenvalues of J1.The eigenspace associated withλ1 is Eλ1(1)=〈(1;0)〉.Note thatw1=(1+α1)[β1-α1/1+α1+d2-γ/1+α1+d1];(1+α1)(β2+1)+α2-δ.The eigenspace associated withλ2 is Eλ21=w1.We get the following:
(a)
Ifα2+(1+α1)β2<δ then λ1<0 and λ2<0; then U1∗ is stable and its stable manifold is Eu=R2.(b)
Ifα2+(1+α1)β2>δ then λ1<0 and λ2>0; then U1∗ is an unstable saddle point of which the stable manifold is Es=〈(1;0)〉 and the unstable manifold is Eu=〈w1〉.(c)
Ifα2+(1+α1)β2=δ then λ1<0 and λ2=0; consequently U1∗ is an equilibrium the stable manifold of which is Es=〈(1;0)〉 and the central manifold is Ec=〈w1〉.(3)
Stability ofU2∗=0;α2-δd2/α2:We have(38)J2=Qδd1α220β2d2δ+α2α2-δα2-δα22-α2-δ.Soλ1=Q(δ)/d1α22 and λ2=-(α2-δ) are the eigenvalues of J2.Let us note thatw2=Q(δ)/d1α22+α2-δ;[β2d2δ+α2(α2-δ)](α2-δ)/α22.The eigenspace associated withλ1 is Eλ1(2)=〈w2〉.The eigenspace associated withλ2 is Eλ2(2)=〈(0,1)〉. We get the following:
(a)
IfQ(δ)<0 then λ1<0 and λ2<0; then, U2∗ is stable and its stable manifold is Es=R2.(b)
IfQ(δ)≥0 then λ1<0 and λ2≥0; then U2∗ is an unstable saddle the unstable manifold of which is Eu=〈w2〉 and the stable manifold of which is Es=〈(0,1)〉.(4)
Stability ofE∗=(u∗;v∗):Letλ1∗ and λ2∗ be the eigenvalues of J. Thus, let us note Tr(J)=λ1∗+λ2∗ its trace and det(J)=λ1∗λ2∗ its determinant. The eigenspace associated with λ1∗ is Eλ1∗=〈J12;λ1∗-J11〉 and the eigenspace associated with λ2∗ is Eλ2∗=〈J21;λ2∗-J22〉.
(a)
Ifdet(J)>0 and Tr(J)<0 then Re[λ1∗]<0 and Re[λ2∗]<0. So, E∗ is stable (stable node or stable focus).(b)
Ifdet(J)≥0 and Tr(J)=0 then λ1∗,λ2∗∈iR. Therefore E∗ is a marginal or a center.(c)
Ifdet(J)<0 or det(J)>0 and Tr(J)>0 then E∗ is unstable. In fact,
(i)
ifdet(J)>0 and Tr(J)>0 then Re[λ1∗]>0 and Re[λ2∗]>0 or λ1∗ and λ2∗ are conjugated complexes. So, E∗ is node or an unstable center,(ii)
ifdet(J)<0 then E∗ is an unstable saddle.


## 5.2. Permanence of Model (9)

Definition 14 (see [8]).
Given solutionu=(x1,x2,…,xn) of a differential system(39)u˙=ft,u,(1)
a componentxi of the solution u of (39) is said to beweakly persistent if limsup[xi(t)]>0,(2)
a componentxi of the solution u of (39) is said to behighly persistent if liminf[xi(t)]>0,(3)
a componentxi of the solution u of (39) is said to beuniformly persistent if there is ε such that liminf[xi(t)]≥ε>0,(4)
System (39) is said to bedissipative as for any component xi of the solution there is a constant Mi>0 such that limsup[xi(t)]≥Mi>0,(5)
System (39) is said to bepermanent if it isuniformly persistent anddissipative.LetΩ¯ be complete metric space and Ω for an open set such that Ω¯=Ω∪∂Ω. Further, we shall take Ω=int(R+2).Definition 15 (see [8]).
A flow or semiflow onΩ¯ under which Ω and ∂Ω are forward invariant is said to be permanent if it is dissipative and if there is a number ε>0 such that any trajectory starting in Ω will be at least at a distance ε from ∂Ω for all sufficiently large t.Definition 16 (see [8]).
(1) Theω-limit set ω(∂Ω) is said to be isolated if it has a covering M=⋃k=1NMk of pairwise disjoint sets Mk which are isolated and invariant with respect to the flow or the semiflow both on ∂Ω and on Ω¯=Ω∪∂Ω.
(2) The setω(∂Ω) is said to be acyclic if there exists an isolated covering M=⋃k=1NMk such that no subset of Mk is a cycle.Lemma 17 (see [8]).
Suppose that a semiflow onΩ¯ leaves both Ω and ∂Ω forward invariant, maps bounded sets in Ω¯ to precompact set for t>0, and it is dissipative. If in addition (1)
ω(∂Ω) is isolated and acyclic,(2)
Es[Mk]∩Ω=∅ for all k∈{1;2;…;N}, where M=⋃k=1NMk is the isolated covering used in the definition of acyclicity of ω(∂Ω) and Es denotes the stable manifold,then the semiflow is permanent.Theorem 18.
Let us assume that0<β1<4 and α2>δ; then we pose (40)Mu=4d2β1+β1d2+2α1β1d2+α12β1d24-β1;L1=4+4-β1Mu2164-β1+Mu+d2β2Mu+1+α2-δ24α2;mu=-β1d2L12-α1d2+γd1L1+1+α1;mv=α2-δd2β2d2+α2;Qx=-d2β1d1x2+γα2d2+d2d1β1α2+d1α1α2x+α22d1-γd2.Let us consider the following assumptions: (41)H1-β1d2L12-α1d2+γd1L1+1+α1>0;H2Qδ>0.Under the assumptions (H1) and (H2), model (9) is permanent and any positive solution (u,v) of (9) verifies(42)mu≤u≤Mu;mv≤v≤L1-mu.Proof.
GivenΩ=int⁡(R+2) and ∂Ω=〈(1;0)〉∪〈(0;1)〉, its frontier, and Ω¯=Ω∪∂Ω, we know that Ω and ∂Ω are invariants for model (9) (cf. Lemma 3) and that A is attractive bounded for any trajectory from Ω¯=R+2 (cf. Theorem 6). Let us assume that 0≤β1<4 and α2>δ and let us apply Lemma 17: (1)
Let us justify the fact that model (9) is dissipative over Ω¯=R+2:
(a)
Let us show thatliminf[u(t)]≥mu=-β1/d2L12-(α1/d2+γ/d1)L1+1+α1. Posing A1=1 and B1=1+α1-L1(β1L1+α1)/d2-γL1/d1, we have du/dt≥(B1-A1u)u and u(0)>0. So, from property (14) of Lemma 4, we have liminf[u(t)]≥mu=-β1/d2L12-(α1/d2+γ/d1)L1+1+α1.(b)
Let us show thatliminf[v(t)]≥mv=(α2-δ)d2/β2d2+α2. Posing B3=(α2-δ) and A3=(β2+α2/d2), then, we have dv/dt≥(B3-A3v)v and v(0)>0. We get liminf[v(t)]≥mv=(α2-δ)d2/β2d2+α2>0 for α2>δ.(c)
We deduce that model (9) is dissipative over Ω¯=R+2 as soon as the assumption (H1) is verified.(2)
Let us prove thatω(∂Ω) is isolated and acyclic. We have ∂Ω=〈(1;0)〉∪〈(0;1)〉. On the one hand, ω[〈(1;0)〉]={U0∗;U1∗}⊂〈(1;0)〉; now the stable manifold U1∗ is Es[U1∗]=〈(1,0)〉 if α2+(1+α1)β2>δ and U0∗ is unstable if α2≥δ. Then any trajectory from 〈(1;0)〉 other than U0∗ approaches U1∗ if α2≥δ for α2≥δ⇒α2+(1+α1)β2>δ. On the other hand, ω[〈(0;1)〉]={U0;U2}⊂〈(0;1)〉; now the stable manifold of U2∗ is Es[U2∗]=〈(0,1)〉 if Q(δ)>0 and U0∗ is unstable if α2≥δ; then any trajectory from 〈(0;1)〉 other than U0∗ approaches U2∗ if α2≥δ and Q(δ)>0. Given that α2>δ then we deduce that ω[∂Ω]={U0;U1;U2}⊂∂Ω is isolated and acyclic if Q(δ)>0 (if the assumption (H2) is verified).(3)
Let us justify thatEs[ω(∂Ω)]∩Ω=∅. We have ω[∂Ω]={U0;U1;U2} and Ω=int⁡(R+2). If α2≥δ, α2+(1+α1)β2>δ, and Q(δ)>0 then Es[ω(∂Ω)]∩Ω=∅; now α2>δ. So Es[ω(∂Ω)]∩Ω=∅ if Q(δ)>0.Definitively, system (9) is permanent.


## 5.3. Limit Cycle of Model (9)

The theorem below presents the conditions for a cyclic growth of the product and the stock of capital of the economy.Theorem 19.
Let us recall the notations of Theorem13. Let us suppose that the assumptions of Theorem 18 are verified and that model (9) admits a unique interior equilibrium E∗=(u∗;v∗). If det⁡(J)>0 and Tr⁡(J)>0 then model (9) admits a limit cycle contained in the attractive region A.Proof.
Under the assumption of Theorem18, model (9) is permanent and if det⁡(J)>0 and Tr⁡(J)>0 then the unique interior equilibrium E∗=(u∗;v∗) is unstable so model (9) admits a limit cycle contained in the compact and bounded region A (from Poincaré-Beddicton’s theorem).


## 6. Global Stability of Model (9)

We now define the conditions in which stability of the product and the stock of capital of the economy are global; that is, they do not depend on the quantities produced and the level of the stock at the initial period. For this study, we define appropriate Lyapunov function.Theorem 20.
PosingMu=4d2β1+(β1d2+2α1)β1d2+α12/β1d2(4-β1) and L1=(4+(4-β1)Mu)2/16(4-β1)+(Mu+d2)(β2Mu+1+α2-δ)2/4α2, let us consider the following assumptions:(43)0<β1<4,α2>δ.(44)(9) admits a single point of interior equilibriumE∗=u∗;v∗,(45)β1L12+α1L1d22+γL1d12<1.(46)0<β1<α1Mu+d2+γMu+d1.Under assumptions (43)–(46), the unique interior equilibrium of model (9) is globally and asymptotically stable.Proof.
Let us consider system (9). Let us suppose that assumption (44) is verified; then, model (9) admits a unique interior equilibrium E∗=(x∗,y∗).
Let us note thatλ=β1(u+d2)/2(β2u+α2). We have V1:R2→R and V2:R2→R such that(47)V1u,v=u-u∗-ln⁡uu∗=∫u∗u1-u∗xdx,V2u,v=λv-v∗-ln⁡vv∗=λ∫v∗v1-v∗xdx.We have the Lyapunov function V:R2→R such that V(u,v)=V1(u,v)+V2(u,v).
Then,dV/dt=dV1/dt+dV2/dt=dV1/duu˙+dV2/dvv˙=(u-u∗)u˙/u+λ(v-v∗)v˙/v. Now (48)u˙u=1-u+β1v+α1-vβ1v+α1u+d2-γvu+d1,v˙v=β2u+α2-δ-β2u+α2u+d2v,and E∗=(u∗;v∗)∈(R+∗)2, a unique interior equilibrium of (9); then, (49)1=u∗-β1v∗+α11-v∗u∗+d2+γv∗u∗+d1;α2-δ=-β2u∗+β2u∗+α2v∗u∗+d2.Therefore,(50)u˙u=u-u∗-1+β1v2+α1vu∗+d2u+d2+γvu∗+d1u+d1+v-v∗β1-β1v+v∗+α1u∗+d2-γu∗+d1;v˙v=u-u∗-β2-β2v∗u∗+d2+β2u+α2vu∗+d2u+d2-β2u+α2u∗+d2v-v∗.Let us pose the following: (51)gu,v=-1+β1v2+α1vu∗+d2u+d2+γvu∗+d1u+d1;hu,v=12β1-β1v+v∗+α1u∗+d2-γu∗+d1+λ2-β2-β2v∗u∗+d2+β2u+α2vu∗+d2u+d2.Consequently, dV/dt=g(u,v)(u-u∗)2+2h(u,v)(u-u∗)(v-v∗)-λ(β2u+α2)/u∗+d2(v-v∗)2. Then, dV/dt≤[g(u,v)+h(u,v)](u-u∗)2+[h(u,v)-λ(β2u+α2)/u∗+d2](v-v∗)2. Thus, dV/dt<0 if g(u,v)<0 and h(u,v)<0, ∀(u,v)∈R2.
Let us determine the conditions on the control parameters of model (9) such that g(u,v)<0 and h(u,v)<0, ∀(u,v)∈R2 (cf. [9], pages 110-111). (1)
Let us overestimateg(u,v). We have g(u,v)=-1+β1v2+α1v/(u∗+d2)(u+d2)+γv/(u∗+d1)(u+d1). Then g(u,v)≤-1+β1L12+α1L1/d22+γL1/d12. Therefore g(u,v)<0, ∀(u,v)∈R2 if -1+β1L12+α1L1/d22+γL1/d12<0.(2)
Let us overestimateh(u,v). We have ∂h/∂v=-(u+d2)β1+λ(β2u+α2)/2(u∗+d2)(u+d2). Now λ=β1(u+d2)/2(β2u+α2); then, ∂h/∂v=-β1/4(u∗+d2)<0. Consequently, h(u,v)≤h(u,0), ∀v≥0. Then, h(u,v)≤1/2[β1-α1/(u∗+d2)-γ/(u∗+d1)]+λ/2[-β2-β2v∗/u∗+d2]. Therefore, h(u,v)≤0, ∀(u,v)∈R2, if β1<α1/Mu+d2+γ/(Mu+d1).(3)
Let us deduce thatdV/dt<0. We know that β1L12+α1L1/d22+γL1/d12<1⇒g(u,v)<0, ∀(u,v)∈R2, and β1<α1/Mu+d2+γ/(Mu+d1)⇒h(u,v)≤0, ∀(u,v)∈R2. Now dV/dt<0, ∀(u,v)∈R2, if g(u,v)<0 and h(u,v)<0, ∀(u,v)∈R2. Then, dV/dt<0, ∀(u,v)∈R2, if assumptions (43)–(46) are verified. Therefore the unique interior equilibrium of model (9) is globally and asymptotically stable if assumptions (43)–(46) are verified.


## 7. Conclusion

Our work used the Kaldor model as basic economic model. By including it at the level of the investment rate and saving rate compatible ecological functions, we encourage the economic actors to adopt a behaviour permitting very rapidly entering a stability area (attracting setA). This stability can be noticed on the one hand in the form of stationary growth of the stock of capital and the product (stable interior equilibrium) and on the other hand in the form of cyclic growth of the capital and the product (limit cycle). We therefore guarantee, under certain conditions, the permanence of the stock of capital, K, and that of the product, Y, in the economy avoiding, in that way, a shortage of the stock of capital or the production in the long term. Under certain conditions, this stability of the financial system (in relation to the capital and the product) is global; that is, it does depend on the level of the stock of capital and the level of production at the initial period.In the first consideration, the model can be applied to a state, regional organisation, or to an enterprise. In the case of an enterprise, the productY refers to the monetary value of the production. We can then infuse the existing production functions such as Cobb-Douglas, Leontief, and CES. In that case, we can substitute the saving with the quantity of work and the tendency will show a technical progress.Secondly, the model can also be applied as an ecological model of two species where one of the species (e.g., man) “cultivate” the other species for its survival or to prevent the loss of that species through a culture rate (investment rate) which is nonnull.



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*Source: 2062819-2017-07-02.xml*