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A class of
n-dimensional ODEs with up to
n feedbacks from the
n’th variable is analysed. The feedbacks are represented by non-specific, bounded, non-negative
C
^{1} functions. The main result is the formulation and proof of an easily applicable criterion for existence of a globally stable fixed point of the system. The proof relies on the contraction mapping theorem. Applications of this type of systems are numerous in biology, e.g., models of the hypothalamic-pituitary-adrenal axis and testosterone secretion. Some results important for modelling are: 1) Existence of an attractive trapping region. This is a bounded set with non-negative elements where solutions cannot escape. All solutions are shown to converge to a “minimal” trapping region. 2) At least one fixed point exists. 3) Sufficient criteria for a unique fixed point are formulated. One case where this is fulfilled is when the feedbacks are negative.

First, an n dimensional system with feedbacks from the n’th variable is introduced and some applications from bio-medicine and biochemistry are described. Then, analysis of a scaled version of the system is made including fixed point investigation. Finally, an easy applicable sufficient criterion for a unique, globally stable fixed point is formulated and proved.

Mathematically, the results in this paper follow from the dimensionless form of the equations stated in (6) of Section 2. But before turning to this form we motivate and discuss the dimensional form of the equations in Section 1 as we relate the system to applications and earlier results.

Many applications of ODEs to physics, chemistry, biology, medicine, and life sciences give rise to non-linear non-negative compartment systems. These include metabolic pathways, membrane transports, pharmacodynamics, epidemiology, ecology, cellular control processes, enzyme synthesis, and control circuits in biochemical pathways [

This paper concerns the stability of the solutions of such models. More specifically, the paper presents criteria for both local and global stability of all systems of ODEs that can be presented as a compartment model with n compartments, on the form shown in

・ Existence of a “trapping region”―a compact set with non negative elements in which any solution will be trapped after finite time.

・ At least one fixed point exists and a real valued function of one variable and the system parameters determines the fixed point.

・ A unique, globally stable fixed point exists if the norm of a real valued function of one variable and the system parameters is less than 1.

A two dimensional model of the HPA axis corresponding to

Some general and analytical considerations partly similar to our has been considered in previous papers [

The mathematical results derived in this article relate to the robustness of hormonal systems, cellular metabolism, etc. The existence of a trapping region ensures that non negative initial (hormone) values lead to (hormone) levels that stay non negative and bounded which is reasonable. Existence of locally stable fixed points may be interpreted as states where (hormone) levels may settle. Perturbing parameters such that a solution enters the basin of attraction to another fixed point may then be interpreted as a new (physiological) state (for a person). Or distinct stable fixed points may be interpreted as states for distinct groups (of people). In case of a unique, globally stable fixed point the long term behaviour is very robust to perturbations.

We consider an n dimensional system of differential equations with n non negative variables

with production rates _{i}’s to be bounded non-negative functions of

back function is the sigmoidal Hill-function

integer. Such Hill-functions are often the result of underlying inter cellular enzymatic reactions regulating feedbacks in the quasi-steady-state approximation [

First a scaling is performed to facilitate the analysis. Defining dimensionless variables

where

Choosing

and

A scaling of Equation (1) thus leads to the dimensionless system

with constants

Since

at least one global solution exists. Here we have made exclusive use of the fact that the

Avoiding negative modelling hormone levels is necessary for a sound model and is proved in the following lemma.

Lemma 1. The non negative hypercube is an invariant solution set to Equation (6)

Proof. Given a solution initially in the non negative hypercube we consider the behaviour at a boundary of the hypercube―a hyperplane defined by

which is non negative for all non negative

which is a product of non negative factors for all non-negative

The fixed point condition of Equation (6) can be expressed

This means that for each fixed point value

may not be explicitly solvable for

Define the functions

and

Thus, finding fixed points of Equation (6) is equivalent to finding

Now choose

Furthermore,

Define the function

Since L and R are continuous so is

We now discuss a sufficient criterion for existence of a unique fixed point of the system. Let

If the feedback functions correspond to negative feedbacks or are independent of

A trapping region is a set,

Lemma 2. Let

and define

Then

Proof.

region’ for

This ensures that

Notice that if

For

For any

Lemma 3. Let

Proof. Follows by the comparison theorem for integrals. W

Lemma 4. Let

Proof. If

Lemma 5. Consider Equation (6). For any

Proof. Fix

Since

Since

Since

U is the ‘minimal’ trapping region. However, if

Fix any

This means H is the restriction of R to

To continue we assume H is positive and a contraction on

For

Thus, two linear systems of differential equations can be constructed with initial condition

and

Then

Since

This means especially

Choosing

since

then

Define

From above there exists a finite time

Now a sequence of sets,

where

and

Lemma 6.

Proof. The proof is done by induction.

Since

are well defined and finite. Then

Since by assumption

and ensures

Due to the squeezing of the solutions using linear systems we have shown that if

We now want to prove that

Lemma 7. Let p be the contraction constant for H. Then

Proof. Follows from the contraction property and the triangle inequality. W

Similarly it follows.

Lemma 8. Let p be the contraction constant for H. Then

Lemma 7 and 8 means we can bound the maximum and minimum of H applied on a compact interval by the maximal distance between any two points in the interval and H evaluated at an end point of the interval.

As mentioned a specific choice of

where

For simplicity we put

Then,

To simplify notation further we introduce

Since

Thus,

For later use we emphasize that

Define

Lemma 9. If H is a contraction and positive on

Proof. The proof is by induction. Since

and

We will show

By inequality (51)

Because

By equality (34)

Using Equation (56)

Then,

Since H is continuous on the compact sets,

Using the contraction property as shown in lemma 7 and lemma 8

and

From the definitions of

and

Thus, we have upper and lower bounds for each of the sets

By definition

and applying Equation (68)

Using Equation (52)

which completes the proof. W

Lemma 10. Let H be defined as in Equation (23). If H is a contraction and positive on

Proof. Fix

Since H is a contraction on a complete metric space the Banach Fixed Point Theorem applies, i.e. a uniquefixed point of

Choose

By Equation (72) there exists

and similarly

There exists a

Inserting from Equations (73)-(76).

Therefore

When

Since

This means that all solutions with initial conditions in

Since all solutions outside

A sufficient, easily applicable criteria for H being a contraction can be formulated [

Lemma 11. Let

If H is positive on

With the results of Section 3 we now have established the main result of global stability of system (6).

Theorem 1. If

The general formulation and results in this paper guarantee that the hormone levels in the models [

A model of mRNA and Hes1 protein production fits

Including time delay in the feedbacks, global stability criteria have been formulated for a subset of possible feedback functions in systems resembling 1 [

A general formulation of an n-dimensional system of differential equations with up to n feedbacks from the n’th variable is formulated. The feedbacks may be non-linear but must be represented by bounded functions which are considered to be the case for some biological systems. Some relevant general results are shown.

・ Existence and uniqueness of solutions are guaranteed.

・ Non-negative initial conditions cause non-negative solutions for all future time.

・ A trapping region,

・ All solutions of the system enter

・ At least one fixed point exists and all fixed points are contained in U. Using

If the feedback functions correspond to negative feedbacks or are independent of

・ If^{1}

Morten Andersen,Frank Vinther,Johnny T. Ottesen, (2016) Global Stability in Dynamical Systems with Multiple Feedback Mechanisms. Advances in Pure Mathematics,06,393-407. doi: 10.4236/apm.2016.65027