Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)

Stability of Solutions in Mixed Differential Equations
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1. DIFFERENCE AND DIFFERENTIAL EQUATIONS

1.1. Tumor Growth Cancer Model

In [1], the authors consider the following system where x describes the density of tumor cells, y the density of hunting predator cells and z the density of resulting cells:

x=1+a1x1xk1xyk2x      y=a2yza3yk3xy                          z=a4z1za5yza6zk4xz(1)
  • a1 is the growth rate of tumor cells

  • a2 represents the conversion rate of the resulting cells to hunting predator cells

  • a3 is the specific loss rate of hunting predator cells

  • a4 represents the growth rate of resting cells

  • a5 is the conversion rate of resting cells to hunting predator cells

  • a6 is the specific loss rate of the resting cells

  • k1 is the rate of killing of tumor cells by hunting cells

  • k2 is the specific loss rate of tumor cells

  • k3 represents the rate of killing of hunting predator cells by tumor cells

  • k4 represents rate of killing of resting cells by tumor cells

The equilibrium points of the system shown in Eq. (1) are:

E00, 0, 0E1x1, 0, 0wherex1=121k2a1+1k2a12+4a1a1>k2E2x2, 0,z2wherex2=121k2a1+1k2a12+4a1andz2=1a6a4k4a4x2a1>k2anda4>a6+k4x2E3x3,y3,z3wherey3=1+a1x31x3k2x3k1x3andz3=a3+k3x3a2a1>k2

The equilibrium point E3x3,y3,z3 is globally asymptotically stable.

Example 1.

For the system shown in Eq. (1) with:

a1=0.6=a4, a2=0.99,a3=0.1,  a5=0.06,a6=0.118k1=0.9,  k2=0.5, k3=0.854,  k4=0.02
we obtain the equilibrium point E31.3213, 0.5656, 0.1186 globally asymptotically stable (Fig. 1).

Figure 1

The equilibrium point E3(1.3213, 0.5656, 0.1186).

1.2. Biolarvicides Against Malaria Model

Biolarvicides are in use in several parts of the world for malaria vector control (see [2]). Consider the system:

x=aβxmIdx+vyy=βxmIvαdymS'=θM1mS+mILθ0+θ1BmSλmsymI'=λmIyθ0+θ1BmIB=γB1Bk+γ1mS+mIB(2)
  • x represents the susceptible humans

  • y represents the infected humans

  • mS represents the susceptible mosquitoes

  • mI represents the infected mosquitoes

  • B represents the biolarvicide population (Fig. 2)

Figure 2

Schematic overview of the system. The direction of each solid line represents movement of population along that line within the same species. The bi-directional dotted lines between boxes indicate a mass-action interaction. The single directional dotted line indicates increase of bacteria population (for example, βxmI is a removal from x population and an addition to y population).

The equilibrium points of the system shown in Eq. (2) are:

  • E0 ad, 0, 0, 0, 0 Disease free, unstable

  • E1 ad, 0, 0, Lθθ0θ, 0 Disease free, unstable

  • E2 ad, 0, 0, 0, K Disease free, unstable if θθ0θ1>k

  • E3 x*,y*, mS*,mI*, 0 Endemic, unstable

  • E4 ad, 0, mS*,0, B* Disease free, stable under conditions

  • E5 x*,y*, mS*,mI*,B* Endemic, stable under conditions

2. DIFFERENCE AND DIFFERENTIAL EQUATIONS WITH DELAYS

2.1. Logistic Equations With Delays

Population density is unlikely to elicit an instant response to the per capita growth rate.

For example, the effect of food scarcity available to young immatures can only be felt later when they reach maturity expressing lower fertility rates. By designating τ the delay interval we get the delayed logistic equation:

Nt= rNt1NtτK
or:
Nt= rNtKNtτK+crNtτ

The earliest delay model in mathematical biology is Hutchinson's equation in 1948 [3], when he modified the classical logistic equation, with a delay term to incorporate hatching and maturation periods into the model and account for oscillations, in the population of Daphnia. Such oscillations (Fig. 3) are in part due to the fact that the fertility of a parthenogenic female is determined, not merely by the population density at a given time, but also by the past densities to which it has been exposed [4].

Figure 3

Oscillations.

2.2. Survival of Blood Cells

The delay differential equation:

Nt=μNt+peγNtτ
has been used by Wazewska-Czyzewska and Lasota [5] as a model for the survival of red blood cells (Fig. 4) in an animal. Here:
  • μ is the probability of death of a red blood cell

  • p and γ are positive constants and are related to the production of red blood cells per unit of time

  • τ is the time required to produce a red blood cell

Figure 4

Red blood cells.

The delay differential equation:

Nt=μNt+peγNtτ
has a unique solution for each initial condition:
Nt=φt          τt0

The positive equilibrium point is given by:

N*=pμeγN*

The solutions oscillate about N* if and only if the equation λ+μ+μγN*eλτ=0 has no real roots.

2.3. Infinite Impulse Response Filter

A filter is a system that functions to extract the data from noise in a signal:

The defining equation for an IIR filter is the difference equation:

yn=i=1n1aiyni+i=0n2bixni
where:
  • xn is the input signal

  • yn is the output signal

  • ai1ini and bi0in2 are real constants

In [6] it is shown that this equation is equivalent to the difference equation:

Δyn+l=0Lclynρl+l=0n1+max1lL ρlpjynj=j=0n2+max1lL ρlqjxnj

3. DIFFERENCE AND DIFFERENTIAL EQUATIONS WITH DELAYS AND ADVANCES

Mixed differential equations have mixed arguments, with delay and advance (Fig. 5). They occur in many problems in economy, biology, physics and engineering. However, this class of equations has been much less studied than other classes of functional differential equations.

Figure 5

Mixed type equations allow to establish conditions between past and potential future events to get a good decision.

Why are these types of equations a challenge? It is well known that the solutions of these types of equations cannot be obtained in closed form [7]. It is not clear how to formulate an initial value problem for such equations and the existence and uniqueness of solutions becomes complicated [8]. To study the oscillation of solutions of differential equations, we need to assume that there exists a solution of such equations on the half line.

Example 2.

[9] Let the initial value problem with t0=0:

xtxht=0,  t0, x0=0

With xt advanced on 0, 2 and delayed on 2, and with:

ht=1                         0t<1t2+4t2   1t<2 tsint2t2

Then:

xt=10t<112t1t<2
is a solution of this initial value problem on 0, 2 which is unbounded on 0, 2 and cannot be extended to 2,.

Example 3.

[9] For α>0, the differential equation:

xt+2αxtα xt+1=0,  t0, x0= 1
has both infinitely growing and decaying solutions eλt on 0,, with λ positive and negative respectively. Indeed, for α=0.25:
xt=e-0.31812txt=e2.4773t

Remark:

Note that for the delayed argument htt and 0α<1, any solution of the equation xt+2αxtα xht=0 tends to zero as t+.

An example with nerve conduction was studied in [10].

The equation:

RC vt = Fvt + v(t  vt+t
where t, v=0 and v+=1, represents a model conduction in a myelinated nerve axon in which the myelin completely insulates the membrane, so that the potential change jumps from node to node (Fig. 6).

Figure 6

Nerve, consisting of axon, myelin sheath and nodes of Ranvier.

In the equation:

RCvt=Fvt+vt +vt+t

vt represents the transmembrane potential at a node and the internodal delay τ represents the reciprocal of the speed of the potential wave as it propagates down the axon.

The constant r is unknown a priori and must be found simultaneously with vt. The constants R and C represent axoplasmic nodal resistivity and nodal capacity, respectively. F includes the model current-voltage relation.

Using the Ohm Law and the Taylor expansion around 0 the equation:

RCvt=Fvt+vt +vt+t
will be transformed at:
vt=a1vt+a2v2t+a3v3t+vtτ2vt+vt+t+Ov4

Using numerical methods, we obtain the solutions in Fig. 7.

Figure 7

Graph of solutions changes when approaching target problem from test problem.

This means the rise time of the membrane potential is faster for lower threshold potential.

Example 4.

The linear autonomous mixed type differential equation:

xt=i=1paixtri+j=1qbjxt+τj
where ai and bj are non-zero real numbers and ri and τj are positive real numbers, can arise in the study of traveling waves in regions with non-local interactions initiated in [11,12].

Example 5.

Stability and determinacy conditions for linear mixed type functional differential equations were studied in [13]:

xt=abxt+θdμθ
where μθ is a real-valued function of bounded variation on a,b.

In this study, the necessary conditions for the existence, uniqueness and stability of a solution to mixed type functional equations were obtained.

4. STABILITY AND SOLUTIONS IN DIFFERENTIAL EQUATIONS WITH DELAYS AND ADVANCES

Consider the differential equation of mixed type:

xt=10xtr1θdvθ+10xt+r2θdηθ
where:
xt
  • r1θ and r2θ are real non-negative continuous functions on 1,0

  • vθ and ηθ are real-valued functions of bounded variation on 1,0

We define:

r1=maxr1θ:  1θ0 r2=maxr2θ:  1θ0

We specify an initial condition of the form:

xt=ϕt         r1tr2
where the initial function ϕ is a given continuous real-valued function on the interval:
r1, r2
satisfying the “consistency condition”:
ϕ0=10ϕr1θdvθ+10ϕr2θdηθ

By a solution of:

xt=10xtr1θdvθ+10xt+r2θdηθ
we mean a continuous function x :r1 , +, which is differentiable on 0 , + and satisfies the equation for every t0 .

If a solution of:

xt=10xtr1θdvθ+10xt+r2θdηθ
is searched in the form xt=eλt for t, the characteristic equation will be:
λ=10eλr1θdvθ+10eλr2θdηθ

The solution of:

xt=10xtr1θdvθ+10xt+r2θdηθ
is said to be stable if for every ε>0 there exists a number =ε>0 such that, for any initial function ϕ with ϕ=maxr1tr2ϕt< , the solution satisfies xt<ε  for all tr1 ,.

Otherwise, the solution is said to be unstable.

The solution is called asymptotically stable if it is stable in the above sense and in addition there exists a number 0>0 such that, for any initial function ϕ with ϕ<0 , the solution satisfies limtxt=0.

5. ESTIMATION OF SOLUTIONS AND STABILITY CRITERIA

Theorem 1.

[14] Let λ0 be a real root of the characteristic equation and:

μλ0=10r1θeλ0r1θdVvθ+10r2θeλ0r2θdVηθ<1βλ0=10r1θeλ0r1θdvθ10r2θeλ0r2θdηθ 

Then the solution x of:

xt=10xtr1θdvθ+10xt+r2θdηθ
satisfies:
xt1+μλ021+βλ0+μλ0Nλ0;ϕeλ0t
with:
Nλ0;ϕ=maxr1tr2eλ0tϕt

Moreover, the solution is:

  • stable if λ0=0

  • asymptotically stable if λ0<0

  • unstable if λ0>0

An important Lemma is the following.

Lemma 1.

[14] Assume that:

10er1θrdvθ+10e r2θrdηθ>1r  ;        10e r1θrdvθ+10er2θrdηθ<1r 
and:
10r1θer1θrdVvθ+10r2θe r2θrdVηθ1 
where r=maxr1,r2.

Then, in the interval 1r, 1r the characteristic equation:

λ=10eλr1θdvθ+10eλr2θdηθ
has a unique root λ0 and this root satisfies the property:
μλ0=10r1θeλ0r1θdVvθ+10r2θeλ0r2θdVηθ<1

Corollary 1.

[14] Assume that:

10er1θrdvθ+10e r2θrdηθ>1r   ,          10e r1θrdvθ+10er2θrdηθ<1r 
and:
10r1θer1θrdVvθ+10r2θe r2θrdVηθ1 
where r=maxr1, r2.

Then the solution of:

xt=10xtr1θdvθ+10xt+r2θdηθ 
is:
  • asymptotically stable if v1+η1>v0+η0

  • unstable if v1+η1<v0+η0

Example 6.

[14] Consider the equation:

xt=10xtθ+12dθ+124+10xt+θ4dη32θ

Notice that in this case we have:

r1θ=θ+12;vθ=θ+124;r2θ=θ4;ηθ=32θ

The characteristic equation is:

λ=10eλθ+12dθ+124+10eλθ4d32θ=1210eλθ+12θ+13e λθ4dθ=1λ 2λ1e λ2e λ2 6e λ41

So, F2λ=λ1λ 2λ1e λ2e λ2 6e λ41 (Fig. 8).

Figure 8

The function F2λ=λ1λ 2λ1e λ2e λ2 6e λ41.

The only one root of F2 is λ0.98.

Then, for λ0=0.98 the condition of Theorem 1 is satisfied. In fact, since v is increasing on 1,0 and  η is decreasing on 1,0:

μλ0=μ0.98max1θ0e0.98θ+12θ+12Vθ+1241,0+max1θ0e0.98θ4θ4V32θ1,0=e0.982214+e 0.9844320.5<1

So, the solution is asymptotically stable.

In this example, stability analysis can be performed using Corollary 1 of Lemma 1 without using the characteristic equation. Indeed, we get:

10eθ+1dθ+124+10eθ2d32θ =1210θ+1eθ+13eθ2dθ0.68>210eθ+1dθ+124+10eθ2d32θ =1210θ+1eθ+13e θ2dθ1.68<210θ+12eθ+1dVθ+124+10θ4e θ2dV32θ e214+e4320.961

Thus, according to Lemma 1, it states that a real root must pass in the interval (−2, 2).

Finally, from Corollary 1 we obtain:

v1+η1=0+32>v0+η0=14+0
and thus the solution is asymptotically stable.

Example 7.

Consider the equation:

xt=10xtθ2dθ4+10xt+θ2dηθ4

Here r1θ=θ2, vθ=θ4, r2θ=θ2, ηθ=θ4 and the characteristic equation is given by:

λ=10eλθ2dθ4+ 10eλθ2dθ4     =1410eλθ2+e λθ2dθ     =12λeλ2e λ2

So, F3λ=λ+12λeλ2e λ2.

The graph of the function F3 (Fig. 9) shows that F3 has two roots: λ0.5 and λ11.

Figure 9

The function F3λ=λ+12λeλ2e λ2.

Let λ=11:

β11=10θ2e 11θ2dθ410θ2e 11θ2dθ42.46μ11

So for λ0=11 Theorem 1 cannot be applied.

Let λ=0.5:

μλ0=μ12=10θ2e θ4dVθ4+10θ2e θ4dVθ4  e14214+e 142140.26<1

Then for λ0=0.5 the conditions of Theorem 1 are satisfied. So the solution is asymptotically stable.

ACKNOWLEDGMENT

Supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT – Fundação Para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020.

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Cite This Article

ris
TY  - CONF
AU  - Sandra Pinelas
PY  - 2023
DA  - 2023/11/29
TI  - Stability of Solutions in Mixed Differential Equations
BT  - Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)
PB  - Athena Publishing
SP  - 71
EP  - 83
SN  - 2949-9429
UR  - https://doi.org/10.55060/s.atmps.231115.007
DO  - https://doi.org/10.55060/s.atmps.231115.007
ID  - Pinelas2023
ER  -
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