Non standard finite difference scheme preserving dynamical properties

Jacky Cressona,b, Frédéric Pierretc,b,∗

aLaboratoire de Mathématiques Appliquées de Pau, Université de Pau et des Pays de l’Adour, avenue del’Université, BP 1155, 64013 Pau Cedex, France

bSYRTE, Observatoire de Paris, CNRS UMR 8630, 77 Avenue Denfert-Rochereau, 75014 Paris, FrancecIMCCE, Observatoire de Paris, CNRS UMR 8028, 77 Avenue Denfert-Rochereau, 75014 Paris, France

Abstract

We study the construction of a non-standard finite differences numerical scheme for a generalclass of two dimensional differential equations including several models in population dynamicsusing the idea of non-local approximation introduced by R. Mickens. We prove the convergenceof the scheme, the unconditional, with respect to the discretization parameter, preservationof the fixed points of the continuous system and the preservation of their stability nature.Several numerical examples are given and comparison with usual numerical scheme (Euler,Runge-Kutta of order 2 or 4) are detailed.

Keywords: Non-standard finite difference methods, qualitative behaviour, qualitativedynamics preserving numerical scheme

1. Introduction

Differentials equations are in general difficult to solve and study. In particular, for most ofthem we do not know explicit solutions. As a consequence, one is lead to perform numericalexperiments using some "integrators" such as the Euler or Runge-Kutta numerical scheme.The construction of these methods is based on approximation theory and focus on the wayto produce finite representation of functions. Although crucial to obtain good agreementsbetween a given solution and its approximation, it is far from being sufficient. Indeed, thesenumerical methods produce artefacts, i.e. numerical behaviour which are not present in thegiven model. Examples of these artefacts are: creation of ghost equilibrium points, change inthe stability nature of existing equilibrium point or destruction of domain invariance, etc.

These issues are of course of fundamental importance and there is a way to solve it. Indeed,the artefacts produced by classical numerical methods are related to the non persistence ofsome important features of the dynamics generated by the differential equation. In particular,

∗Corresponding authorEmail addresses: jacky.cresson@univ-pau.fr (Jacky Cresson), frederic.pierret@obspm.fr (Frédéric

Pierret)

Preprint submitted to Elsevier December 16, 2015

the qualitative theory of differential equations is mainly concerned with invariant objects suchas equilibrium points and their dynamical properties such as stability or instability as well asother global properties such as domain invariance, variational structures, etc. Therefore, anidea emerged to construct numerical schemes that do not focus on the approximation prob-lems but deal with some dynamical informations leading to what can be called qualitativedynamical numerical scheme.

This program was in fact mainly developed by R. Mickens in a serie of papers (see [17, 18, 19]).In order to distinguish the new numerical scheme from the classical one, he coined the termnonstandard schemes for them.

The aim of this paper is to introduce a nonstandard scheme concerning a class of differentialequations that include all prey-predator models. The study of nonstandard scheme for prey-predator models is extensive but has been done only with specific form of the differentialequations (see [10, 11, 12]). Our results generalize the one obtained by D.T. Dimitrov andH.V. Kojouharov in [10] and related works discussed in [3, 4, 13].Precisely, we give a complete answer to the following questions and problems:

• Is the non-standard scheme convergent?

• For which value of the time step increment, do we recover the stability of the fixedpoints?

• To give a comparison between the numerical results obtained using the non-standardscheme and other well known methods (such as Euler, Runge-Kutta 2 or Runge-Kutta4).

To our knowledge, these questions and problems are not solved in the previous cited paper. Inthis article, we prove convergence of the non-standard scheme which is constructed. A com-parison between our scheme and Euler, Runge-Kutta 2 and 4 is also given. Precisely, classicalproblems related to the behaviour of these schemes with respect to equilibrium points, stabil-ity and positivity are discussed with respect to the time step increment. Such a comparisonis not provided in the existing literature. Moreover, we prove that two of the third class ofequilibrium points are preserved unconditionally with respect to the time-step increment. Incomparison, in [3, 4, 13], the authors obtain only stability for a sufficiently small time-stepwhich in fact follows directly from standard arguments in dynamical systems theory (see Sec-tion 6.4).

The plan of the paper is as follows:

2

In Section 2, we recall classical definitions of equilibrium points and their stability for discreteand continuous dynamical systems. Section 3 gives the definition of a non-standard finitedifference scheme following R. Anguelov and J.M-S. Lubuma ([1, 2]). In Section 4, we introducethe class of two dimensional differential equations that we are considering and we study thepositivity and the stability of the equilibrium points of this class of differential equations. InSection 5, we introduce the non-standard scheme associate to this system with results aboutthe preservation of stability and positivity of the initial problem. In Section 6, we illustratenumerically the results on different models. Section 7 concludes the paper and provides someperspectives and comments.

2. Reminder about continuous/discrete dynamical systems

In this section, we remind classical results about continuous and discrete dynamical systemsdealing with the qualitative behaviour of ordinary differential equations which will be studiedboth for our class of models and their discretization. We refer in particular to the book of S.Wiggins [24] for more details and proofs.

2.1. Vector fields2.1.1. Equilibrium points and stability

Consider a general autonomous differential equation

dx(t)

dt= f(x(t)), x ∈ Rn, (1)

where f ∈ C2(Rm,Rm) is called the vector fields associated to (1).

An equilibrium solution of (1) is a point E ∈ Rn such that f(E) = 0. We denote by F the setof equilibrium points of (1).

An important issue is to understand the dynamics of trajectories in the neighborhood of agiven equilibrium point. This is done through different notions of stability. In our model, wewill use mainly the notion of asymptotic stability which is a stronger notion than the usualLiapounov stability.

Definition 2.1 (Liapounov stability). A solution x(t) of (1) is said to be stable if, givenε > 0, there exists δ = δ(ε) > 0 such that, for any other solution, y(t), of (1) satisfying‖ x(t0)− y(t0) ‖< δ, then ‖ x(t)− y(t) ‖< ε for t > t0, t0 ∈ R.

Our main concern will be asymptotic stability.

Definition 2.2 (Asymptotic stability). A solution x(t) of (1) is said to be asymptoticallystable if it is Liapounov stable and for any other solution, y(t), of (1), there exists a constantδ > 0 such that if ‖ x(t0)− y(t0) ‖< δ, then lim

t→+∞‖ x(t)− y(t) ‖= 0.

3

For an equilibrium E, an important result is that asymptotic stability can be determined fromthe associated linear system defined by

dy

dt= Df(E)y, (2)

where Df(E) is the Jacobian of f evaluated at point E.

Precisely, we have (see [24], Theorem 1.2.5 p.11):

Theorem 2.3. Let E be an equilibrium point of (1). Assume that all the eigenvalues of Df(E)

have negative real parts. Then the equilibrium point E is asymptotically stable.

2.1.2. Positivity invariance

In many applications, in particular Biology, the variables representing the dynamical evolutionof the system must belong to a given domain. A classical example is given by variablesassociated to density of population which must stay positive during the time evolution. Sucha constraint is called positivity and is defined as follows.

Definition 2.4. The system (1) satisfies the positivity property if for all initial conditionx0 ∈ (R+)m and initial time t0 ∈ R+ we have x(t) ∈ (R+)m for all t ≥ t0.

The positivity property can be tested using the following necessary and sufficient condition(see [23] and [21]):

Theorem 2.5. The set

K+ := {x = (x1, . . . , xm) ∈ Rm, xi ≥ 0, i ∈ I}

is invariant for (1) if and only if

fi(x) ≥ 0 for x ∈ K+ such that xi = 0,

for all i ∈ I.

2.2. Maps

Numerical scheme defines maps which can be studied as discrete dynamical systems.

2.2.1. Fixed points and stability

Consider a Cr (r ≥ 1) mapx 7→ φ(x), x ∈ Rn. (3)

The map φ induces a discrete dynamical system defined by

xn+1 = φ(xn), xn ∈ Rn. (4)

4

Let x0 ∈ Rn be given. We denote by φn = φ ◦ · · · ◦ φ n-times. The bi-infinite sequence{φn(x0), n ∈ Z} is called the orbit of x0 under the map φ.

Everything discuss for vector fields possesses a discrete analogue. In particular, equilibriumpoint for vector fields correspond to fixed points for maps, i.e. point E such that φ(E) = E.We denote by F the set of fixed points of (4).

Theorem 2.6. Let E be a fixed point of (4). Assume that all the eigenvalues of the Jacobianmatrix Dφ(E) have moduli strictly less than one. Then the fixed point E is asymptoticallystable.

2.2.2. Positivity invariance

The positivity invariance for vector fields has also an analogue in the discrete setting:

Definition 2.7. The discrete dynamical system (4) satisfies the positivity property if for allinitial conditions x0 ∈ (R+)n, we have xn ∈ (R+)n for all n ≥ 0.

A necessary and sufficient condition for positivity is that φ(x0) ≥ 0 for all x0 ≥ 0. Althoughsimple, this condition is in general difficult to check for a given map.

3. Reminder about non standard numerical scheme

We suppose the whole integration occurs over an interval [t0, T ] with T ∈ R+. Let h ∈ R withh > 0. For k ∈ N, we denote by tk the discrete time defined by tk = kh.

Definition 3.1. A general one-step numerical scheme with a step size h, that approximatesthe solution of a general system such as (1) can be written in the form

Xk+1 = φh(Xk) (5)

where φh is C2(Rm,Rm) and Xk is the approximate solution of (1) at time tk, for all k ≥ 0

and X0 the initial value.

Definition 3.2. A numerical method converges if the numerical solution Xk satisfies

sup0≤tk≤T

‖Xk − x(tk)‖∞ → 0 (6)

as h→ 0 and X0 → x(t0).It is accurate of order p if

sup0≤tk≤T

‖Xk − x(tk)‖∞ = O(hp) +O(‖X0 − x(t0)‖∞) (7)

as h→ 0 and X0 → x(t0).

5

Following R. Anguelov and J.M-S. Lubuma (see [1, 2]), we define the notion of non-standardfinite difference scheme as follows:

Definition 3.3. A general one-step numerical scheme that approximate the solution of (1)is called Non-Standard Finite Difference scheme if at least one of the following conditions issatisfied:

• ddtX(tk) is approximate as Xk+1−Xk

ϕ(h) where ϕ(h) = h+O(h2) a nonnegative function,

• φh(f,Xk) = φ̃h(f,Xk, Xk+1) is a nonlocal approximation of f(tk, X(tk)).

The terminology of nonlocal approximation comes from the fact that the approximation of agiven function f is not only given at point xk by f(xk) but can eventually depend on morepoints of the orbits as for example

x2(tk) ≈ xkxk+1, xkxk−1, xk

(xk−1 + xk+1

2

),

x3(tk) ≈ x2kxk+1, xk−1xkxk+1.

In the previous definition, we have concentrated on the simpler case, depending only on xk

and xk+1.

4. A class of ordinary differential equations

We consider the two dimensional system of ordinary differential equations defined for (x, y) ∈R× R by

dx

dt= x (f+(x, y)− f−(x, y)) , x(t0) = x0 ≥ 0,

dy

dt= y (g+(x, y)− g−(x, y)) , y(t0) = y0 ≥ 0,

(E)

where f+, f− and g+, g− are positive for all (x, y) ∈ R+ × R+ and of class C1.

The vector field associated to (E), denoted by ϕ : R2 → R2, is defined by

ϕ(x, y) =

(x (f+(x, y)− f−(x, y))y (g+(x, y)− g−(x, y))

)(8)

Equation (E) contains classical examples such as the general Rosenzweig-MacArthur predator-prey model (see [5], p. 182) studied in particular by D. T. Dimitrov and H. V. Kojouharov[10].

6

4.1. Equilibrium points and stability

The set of equilibrium points of (E) is denoted by F . By definition, a point (x, y) ∈ F satisfies

x (f+(x, y)− f−(x, y)) = 0,y (g+(x, y)− g−(x, y)) = 0.

(9)

Equilibrium points of (E) consists of the origin O = (0, 0) and potential equilibrium pointswhich can belong to three distinct families given by

E1 = (x], 0) where x] 6= 0 and f+(x], 0) = f−(x], 0) ,

E2 = (0, y]) where y] 6= 0 and g+(0, y]) = g−(0, y])

E3 = (x?, y?) with x? 6= 0, y? 6= 0 where f+(x?, y?) = f−(x?, y?) and g+(x?, y?) = g−(x?, y?) ,

depending on the existence of solutions for each equation. The family E1 and E2 can naturallybe included in the family E3 if we allow null components. However, in many examples, onlyfamily E1 and E2 appear. Moreover, the preservation of a point of the family E3 behaves ingeneral very differently as the preservation of an equilibrium point of the families E1 and E2

(see Section 6.3).

The stability/instability nature of these equilibrium points can be completely solved. Indeed,we have the following Lemmas describing explicitly the eigenvalues for each type of equilibriumpoint.

Lemma 4.1. The origin has eigenvalues given by λ01 = (f+ − f−) (0, 0) and λ02 = (g+ − g−) (0, 0).

The proof is given in Appendix A.1.

Lemma 4.2. Assume that (E) possesses an equilibrium point belonging to the family E1 (resp.E2) denoted by (x], 0) (resp. (0, y])). The eigenvalues are given by λ11 = x] (∂xf+ − ∂xf−) (x], 0)

and λ12 = (g+ − g−) (x], 0) (resp. λ21 = f+(0, y])−f−(0, y]) and λ22 = y] (∂yg+ − ∂yg−) (0, y])).

The proof is given in Appendix A.2.

Finally, we have the following general result:

Lemma 4.3. Assume that (E) possesses an equilibrium point belonging to the family E3

denoted by (x∗, y∗). We denote by T (resp. D) the trace (resp. determinant) of the Jacobianmatrix of φ at point (x∗, y∗) denoted by Dϕ(x?, y?) is given by

T = x? (∂xf+ − ∂xf−) (x?, y?) + y? (∂yg+ − ∂yg−) (x?, y?), (10)

7

and

D = x?y? ((∂xf+ − ∂xf−) (∂yg+ − ∂yg−)− (∂yf+ − ∂yf−) (∂xg+ − ∂xg−)) (x?, y?). (11)

If T 2 − 4D ≥ 0 then Dϕ(x?, y?) has eigenvalues given by1

2(T ±

√T 2 − 4D).

Else if T 2 − 4D < 0 then Dϕ(x?, y?) has eigenvalues1

2(T ± i

√4D − T 2) with i2 = 1.

Proof. The Jacobian of Φ at an equilibrium point of the family E3 is given by

Dϕ(x?, y?) =

(x? (∂xf+ − ∂xf−) (x?, y?) x? (∂yf+ − ∂yf−) (x?, y?)y? (∂xg+ − ∂xg−) (x?, y?) y? (∂yg+ − ∂yg−) (x?, y?)

). (12)

The characteristic polynomial is then given by λ2 − Tλ + D = 0 where T and D correspondto the trace and determinant of Dϕ(x?, y?). This concludes the proof.

We denote by (SEi) for i = 0, 1, 2, 3 the conditions where Re(λi1) and Re(λi2) are strictlynegatives that is to say (SEi) is the conditions for which the equilibrium point in Ei is linearlyasymptotically stable (and then asymptotically stable by Theorem 2.3). Using the previ-ous Lemmas we have the following explicit characterization of linearly asymptotically stableequilibrium points in each family:

Lemma 4.4 (Conditions of linear asymptotic stability). The conditions of asymptotic stability(SEi) for i = 0, 1, 2, 3 are given by:

• The origin is linearly asymptotically stable if and only if (f+ − f−) (0, 0) < 0 and(g+ − g−) (0, 0) < 0.

• An equilibrium point belonging to the family E1 (resp. E2) denoted by (x], 0) (resp.(0, y])) is linearly asymptotically stable if x] (∂xf+ − ∂xf−) (x], 0) < 0 and (g+ − g−) (x], 0) <

0 (resp. f+(0, y])− f−(0, y]) < 0 and y] (∂yg+ − ∂yg−) (0, y]) < 0).

• An equilibrium point of the family E3 is linearly asymptotically stable if and only if T < 0

and D > 0.

These conditions will be used in Section 6.3. Only the third condition is not trivial althoughclassical. It uses the trace-determinant diagram to characterize the dynamical behaviour oflinear systems (see [16]).

4.2. Positivity invariance

Using Theorem 2.5, we easily derive the following result:

Theorem 4.5. The system (E) satisfies the positivity property.

Proof. The conditions of Theorem 2.5 are clearly satisfied for (E).

8

5. A non-standard finite difference scheme

The notion of non-standard scheme was introduced by R. E. Mickens at the end of the 80’s.We refer to the book [17] in particular Chapter 3 for more details and an overview of Mickens’sideas and to [20] for a more recent presentation.

5.1. Definition

We introduce the following non-standard finite difference scheme:

Definition 5.1. The NSFD scheme of (E) is given by

xk+1 − xkh

= xkf+(xk, yk)− xk+1f−(xk, yk),

yk+1 − ykh

= ykg+(xk, yk)− yk+1g−(xk, yk).(13)

The associated discrete dynamical system is defined by the map ϕNS,h : R2 → R2 given by

ϕNS,h(xk, yk) =

xk(

1 + hf+(xk, yk)

1 + hf−(xk, yk)

)yk

(1 + hg+(xk, yk)

1 + hg−(xk, yk)

) (14)

As usual, the main issue for numerical scheme is to prove convergence. We have the followingresult:

Theorem 5.2. The NSFD scheme (13) is convergent and of order one.

The proof is given in Appendix B.

6. Dynamical properties of the NSFD scheme

6.1. Positivity invariance

As f+, f− and g+, g− are positive for all (x, y) ∈ R+ × R+, we have:

Lemma 6.1. The NSFD scheme preserves positivity for arbitrary h.

6.2. Equilibrium points

In general we have F ⊂ Fh because numerical schemes induce sometimes artificial fixed pointssuch as the Runge-Kutta methods. These points are often called extraneous or ghost fixedpoints (see [6] p. 16). The NSFD scheme behaves very nicely:

Lemma 6.2. For arbitrary h, we have F = Fh.

9

Proof. Fixed points of the mapping ϕNS,h are given by ϕNS,h(x, y) = (x, y) which is equivalentto

x

(1 + hf+(x, y)

1 + hf−(x, y)

)= x,

y

(1 + hg+(x, y)

1 + hg−(x, y)

)= y,

which reduces to {x = 0 or f+(x, y) = f−(x, y),

y = 0 or g+(x, y) = g−(x, y).

These conditions are clearly equivalent to conditions (9) for the vector field ϕ. Then, (x, y)

is a fixed point for the mapping ϕNS,h if and only if (x, y) is an equilibrium point for ϕ. Inother words, we have F = Fh which concludes the proof.

Remark 6.3. This property, although essential, is of course also satisfied by the most simpleEuler method which is an effect of one step numerical scheme and the fact that these methodsare of order one. In the contrary, the Runge-Kutta methods of order 2 and 4, do not preservethe set of equilibrium points due to non trivial solutions to the fixed point equation whichproduce ghost equilibrium points (see Section 7.1).

6.3. Stability and instability

The stability of equilibrium point of (E) under discretization will correspond to the stabilityof the fixed point of the map ϕNS,h. We have:

Theorem 6.4. The NSFD scheme preserves the stability nature of the origin and equilibriumpoints of type E1 or E2 for arbitrary h.

The proof is given in Appendix C.

For equilibrium points belonging to the family E3, we have not preservation of the stabilitynature unconditionally with respect to the parameter h but only for a sufficiently small one.

Theorem 6.5. If (SE3) are satisfied then there exist a constant C(E3) which can be computedexplicitly such that for all 0 < h < C(E3), E3 is a stable fixed point for ϕNS,h.

The proof is given in Appendix D.

6.4. A remark concerning preservation of stability by numerical scheme

In [13, Theorem 3.1], the authors prove a result which can be reformulated as follows: undersome conditions ensuring that the equilibrium points of the differential equation is stable, theNSFD numerical scheme preserves the stability of the associated fixed point for a sufficientlysmall step-size h.

10

It must be noted that this result is again not specific to the non standard case. It can beformulated for the Euler scheme as well. More generally, it can be formulated for any numer-ical scheme with a minor modification. Indeed, in general, one can not ensure that the stablefixed point of the numerical scheme and the equilibrium points of the differential equationcoincide but only that it exists in a sufficiently small neighborhood. Moreover, this result isnot related to the stable character of the equilibrium points but more generally to the factthat these points are hyperbolic (see [24, Definition 1.2.6 p.12]).

There exists at least two ways to prove this result. One using vector field and another one,strictly equivalent, dealing only with mappings.

More precisely, let us consider a hyperbolic equilibrium point of a differential equation (let sayin dimension two for this example), i.e. an equilibrium point such that the linearized systemat this point possesses only eigenvalues with non zero real part. Hyperbolic equilibrium pointsare structurally stable (see [24, Example 12.1.1 p.163]), i.e. that they persist under any smallperturbation of the vector field f in C1 topology. This result is a consequence of the invariantmanifold theorem proved in [15, Theorem 1.1].

Moreover, in [14, Section IX.1, Theorem 1.2], the authors prove that a general numericalscheme ϕh can be seen as the time-one map of the flow associated to a modified vector field.More precisely, there exists a vector field fh that is C1 close to f , when h is sufficiently small,and such that its flow ϕh,t satisfies

ϕh,1 = ϕh. (15)

Combining this result and the structural stability of hyperbolic fixed points, we deduce thathyperbolic fixed points are preserved by any numerical scheme if the time step increment issufficiently small.

As a consequence, the only non trivial part in such kind of result is to give some informationabout the threshold under which the persistence is effective. In that respect, the theoremsproved in [13] concerning equilibrium points and stability can be recovered directly fromthe previous argument only knowing that the fixed points for the numerical scheme and theequilibrium points of the equations coincide.

Remark 6.6. Of course, the previous result does not hold in general for non hyperbolic equi-librium points, in particular for center equilibrium (see [24, Section 1.4 p.16]) which are ubiq-uitous in many applications.

It must be note that the unconditional preservation of the stability of the fixed points can not

11

be recovered by such kind of general argument.

6.5. A remark concerning non-locality and weighted time step

R. Mickens has derived many "tricks" in order to preserve particular dynamical behaviour.The one used in this paper is a non-local approximation of a given function. There existsalso the possibility to use a weighted time step. This is done for example in [10] where theauthors mix the two tricks in order to preserve the stability/positivity for a particular case ofour model. This approach gives in our case the following numerical scheme:

Definition 6.7. Let ϕ(h) = h+O(h2) be a nonnegative function. The extended NSFD schemeof (E) is given by

xk+1 − xkϕ(h)

= xkf+(xk, yk)− xk+1f−(xk, yk),

yk+1 − ykϕ(h)

= ykg+(xk, yk)− yk+1g−(xk, yk).

This scheme defines a natural map

ϕENS,h(xk, yk) =

xk ( 1+ϕ(h)f+(xk,yk)1+ϕ(h)f−(xk,yk)

)yk

(1+ϕ(h)g+(xk,yk)1+ϕ(h)g−(xk,yk)

)Our results can be extended to this new numerical scheme. However, the main propertieshave nothing to do with the choice of a weighted time increment but are only induced by thenon-local approximation.

7. Numerical examples

Our aim in this Section is to illustrate the advantages of the non-standard scheme with respectto other classical methods including the Euler scheme or the Runge-Kutta method of order 2or 4 on a specific example. In particular, we provide simulations illustrating some well-knownnumerical artefacts produced by these methods and which are corrected by the non-standardscheme.

We consider the following class of model:

dx

dt= x

(b−

(bx+ ay

c+x

)),

dy

dt= y

(xc+x − d

),

where a, b, c, d are real constants.

We use two particular sets of values for our simulations:

• Model 1: a = 2, b = 1, c = 0.5, d = 6.

• Model 2: a = 2, b = 1, c = 1, d = 0.2.

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7.1. Equilibrium points artefacts

Model 1 possesses two equilibrium points corresponding to the origin with eigenvalues λ01 = 1,λ02 = −6 and an equilibrium point of type E1 given by P1 = (1, 0) with eigenvalues λ11 = −1,λ12 = −16

3 .

In figure 1, we provide numerical simulations for the initial conditions x0 = 15, y0 = 0.1. Themain result is that the NSFD scheme has a better dynamical behaviour than the Runge-Kuttamethod of order 2. In particular the Runge-Kutta method of order 2 produces for h = 0.1 avirtual equilibrium point.

0

0.02

0.04

0.06

0.08

0.1

0 2 4 6 8 10 12 14 16

NSFDEULER

RK2RK4

(a) h = 0.01

0

0.02

0.04

0.06

0.08

0.1

0 2 4 6 8 10 12 14 16

NSFDEULER

RK2RK4

(b) h = 0.1

0

0.02

0.04

0.06

0.08

0.1

0 2 4 6 8 10 12 14 16

NSFDEULER

RK2RK4

(c) h = 0.2

Figure 1: Numerical simulations of Example 1 with x0 = 15, y0 = 0.1.

Moreover, the NSFD scheme behaves equivalently to the Runge-Kutta method of order 4.

13

From the computational point of view, this result is very substantial since the algorithmiccomplexity of the NSFD is much lower than that of the Runge-Kutta of order 4.

7.2. Stability/instability artefacts

We use again Model 1. The simulations are made with the initial condition x0 = 0.3, y0 = 7.5

and are given in Figure 2.The NSFD scheme reproduces the correct dynamical behaviour already for h = 0.5. In thecontrary, the Euler, Runge-Kutta of order 2 or 4 do not match the real dynamics for h fromh = 0.5 to h = 0.3. The Runge-Kutta of order 4 produces a better agreement for h = 0.2

but with artificial oscillations. The correct behaviour is only recovered for h = 0.1 for theRunge-Kutta of order 4 and h = 0.01 for the others.

Another example with simulations done with initial conditions x0 = 0.4, y0 = 0.4 is given inFigure 3.The NSFD scheme reproduces the correct dynamical behaviour already for h = 0.6. The Eu-ler, Runge-Kutta of order 2 does not match the real dynamics for h from h = 0.6 to h = 0.2.The Runge-Kutta of order 4 produces a better agreement for h = 0.4 but with a completelydifferent trajectory for h = 0.5 even if the convergence to the equilibrium point P1 is respected.The correct behaviour is only recovered for h = 0.1 for the others.

Model 2 possesses three equilibrium points corresponding to the origin with eigenvalues λ01 = 1,λ02 = −1

5 , and one fixed point in the family E1 and E3: P1 = (1, 0) with eigenvalues λ11 = −1,λ12 = 3

10 and P3 = (14 ,1532) with eigenvalues λ31 = 1

20

(−1 + i

√47), λ32 = 1

20

(−1− i

√47).

The equilibrium point P3 is stable. Theorem 6.5 ensures that the NSFDM scheme preservesthe stability as long as 0 < h < C = 1. However, the benefit of using the NSFD scheme is notas evident as in the previous case. Indeed, as displays in Figure 4 only the Runge-Kutta oforder 4 converges to the equilibrium point for h from h = 4 to h = 0.01. For h between h = 4

and h = 1 the NSFDM has a periodic limit cycle. The Runge-Kutta of order 2 converge tothe equilibrium point from h = 2 to h = 0.01 but for h = 4 it diverges. The Euler has also aperiodic limit cycle limit for h from h = 2 to h = 1.

7.3. Invariance/positivity artefacts

By construction, the NSFD respects the positivity of the systems unconditionally with respectto the time increment h by Lemma 6.1. However, this is not the case for the classical numericalscheme:

• Figure 1 shows that the Euler and the Runge-Kutta methods of order 2 or 4 do notrespect the positivity property.

14

• Figure 2 shows that the Euler method does not respect the positivity property for hfrom h = 0.5 to h = 0.1. In the same way, the Runge-Kutta of order 4 does not respectpositivity for h from h = 0.5 to h = 0.2.

8. Conclusion

The nonstandard scheme studied in this paper generalizes previous results obtained by D.T.Dimitrov and H.V. Kojouharov in a series of papers [10], [11] and [12]. The convergence isproved as well as the fact that the scheme preserves the fixed points and their stability natureand also the positivity. The main advantages of this scheme are illustrated via numericalexamples. These simulations show at least two things:

• First, most of the time the non-standard scheme behaves better or equivalently to aRunge-Kutta method of order 4. The algorithmic complexity of the non-standard schemebeing comparable to the Euler scheme, the gain in term of computation is considerable.

• Second, the preservation of the dynamical constraints leads to a scheme which gives thegood dynamical behaviour even for large time increment. This gives also a very bigimplementation and computational advantage.

The positivity or more generally domain invariance is an important issue for many appli-cations, in particular in biology, and provide the first test to select models. This questionis fundamental when one is dealing with stochastic differential equations (see [8] and [7])as simulations are used to validate a given model. An interesting issue is then to developstochastic qualitative dynamical numerical scheme for stochastic differential equations. A firststep in this direction has been made by F. Pierret [22] by the construction of a non-standardEuler-Murayama scheme.

9. Acknowledgments

We would like to thank the referees for pointing us some references related to our work [3, 4, 13].

Appendix A. Proof of Lemmas 4.1 and 4.2

Appendix A.1. Proof of Lemma 4.1

The Jacobian matrix of φ is given by

Dϕ(x, y) =

((f+ − f−) (x, y) + x (∂xf+ − ∂xf−) (x, y) x (∂yf+ − ∂yf−) (x, y)

y (∂xg+ − ∂xg−) (x, y) (g+ − g−) (x, y) + y (∂yg+ − ∂yg−) (x, y)

).

(A.1)At the origin the Jacobian matrix reduces to

Dϕ(0, 0) =

((f+ − f−) (0, 0) 0)

0 (g+ − g−) (0, 0)

).

The eigenvalues are then given by λ01 = (f+ − f−) (0, 0) and λ02 = (g+ − g−) (0, 0).

15

Appendix A.2. Proof of Lemma 4.2

Using equation (A.1), we deduce that the Jacobian of φ at an equilibrium point of the familyE1 is given by

Dϕ(x], 0) =

(x] (∂xf+ − ∂xf−) (x], 0) x] (∂yf+ − ∂yf−) (x], 0)

0 (g+ − g−) (x], 0)

). (A.2)

The eigenvalues are easily obtained as λ11 = x] (∂xf+ − ∂xf−) (x], 0) and λ12 = (g+ − g−) (x], 0).

In the same way, the Jacobian at an equilibrium point of the family E2 has eigenvaluesλ21 = f+(0, y])− f−(0, y]) and λ22 = y] (∂yg+ − ∂yg−) (0, y]).

Appendix B. Proof of Theorem 5.2

As usual in the study of numerical algorithm (see [9],Chap.VIII,p.226-228), we prove consis-tency and stability of the NSFD scheme and then convergence (see [9],Corollaire,p.227).

The NSFD sheme is given by

xk+1 = xk

(1 + hf+(xk, yk)

1 + hf−(xk, yk)

),

yk+1 = yk

(1 + hg+(xk, yk)

1 + hg−(xk, yk)

).

A Taylor expansion with remainder of each component gives

x(tk+1) = x(tk) + hdx

dt(tk) +

1

2h2d2x

dt2(tk + θxh) = x(tk) + hϕ1(x(tk), y(tk)) + τ1k

y(tk+1) = y(tk) + hdy

dt(tk) +

1

2h2d2y

dt2(tk + θyh) = y(tk) + hϕ2(x(tk), y(tk)) + τ2k

for some real θx and θy between 0 and 1. This defines the local truncation error τk.Let Xk = (xk, yk) and X(tk) = (x(tk), y(tk)) for all k ≥ 0. Subtraction of the NSFD schemeand Taylor expansions gives a difference equation for the error

ek+1 = Xk+1 −X(tk+1) = ek + h

xk (f+−f−1+hf−

)(Xk)− x(tk) (f+ − f−) (X(tk))

yk

(g+−g−1+hg−

)(Xk)− y(tk) (g+ − g−) (X(tk))

− τnBy hypothesis the first component of ϕ is Lipschitz with constant L1 and the second compo-nent is also Lipschitz with constant L2. We denote by L the maximum value between L1 andL2.

16

As 1+hf−(x, y) ≥ 1 and 1+hg−(x, y) ≥ 1 for all x, y ≥ 0 and assume that the local truncationerror satisfies a bound ‖τk‖∞ ≤ τ for all k then

‖ek+1‖∞ ≤ ‖ek‖∞ + hL‖ek‖∞ + τ. (B.1)

Using the discrete Gronwall Lemma (see [9],p.333) we obtain

‖ek‖∞ ≤ eLT ‖e0‖∞ +eLT − 1

LTkτ. (B.2)

We then have stability.

The local truncation error is equal to

‖τk‖∞ = sup

(∣∣∣∣12h2d2xdt2 (tk + θxh)

∣∣∣∣ , ∣∣∣∣12h2d2ydt2 (tk + θyh)

∣∣∣∣) ≤ 1

2Mh2. (B.3)

As ϕ is C1([t0, T ],R2), we deduce that the solution is C2([t0, T ],R2) with bounded derivativesover [t0, T ]. As a consequence, we obtain

‖τk‖∞ ≤1

2Mh2. (B.4)

We deduce that the NFSD scheme is consistent.

Consistency gives a local bound on τ and stability allows us to conclude convergence:

‖Xk −X(tk)‖∞ ≤ eLT ‖X0 −X(t0)‖+eLT − 1

LT

T

2Mh ≤ O(‖X0 −X(t0)‖∞) +O(h)

Appendix C. Proof of Theorem 6.4

The Jacobian matrix is given by

DϕNS,h(x, y) = 1+hf+(x,y)1+hf−(x,y) + hx

((1+hf−)∂xf+−(1+hf+)∂xf−

(1+hf−)2

)(x, y) hx

((1+hf−)∂yf+−(1+hf+)∂yf−

(1+hf−)2

)(x, y)

hy((1+hg−)∂xg+−(1+hg+)∂xf−

(1+hg−)2

)(x, y) 1+hg+(x,y)

1+hg−(x,y) + hy((1+hg−)∂yg+−(1+hg+)∂yf−

(1+hg−)2

)(x, y)

.

At the origin, the Jacobian reduces to

DϕNS,h(0, 0) =

(1+hf+(0,0)1+hf−(0,0) 0

0 1+hg+(0,0)1+hg−(0,0)

)

has eigenvalues

17

γ01 =1 + hf+(0, 0)

1 + hf−(0, 0)and γ02 =

1 + hg+(0, 0)

1 + hg−(0, 0).

The stability conditions (SE0) imply f+(0, 0) < f−(0, 0) and g+(0, 0) < g−(0, 0) then

0 <1 + hf+(0, 0)

1 + hf−(0, 0)< 1, 0 <

1 + hg+(0, 0)

1 + hg−(0, 0)< 1,

for arbitrary h. The origin is then stable for arbitrary h.

At E1 the Jacobian

DϕNS,h(x], 0) =

1 + hx]

(∂xf+−∂xf−

1+hf+

)(x], 0) hx]

(∂yf+−∂yf−

1+hf+

)(x], 0)

01+hg+(x],0)1+hg−(x],0)

has eigenvalues

γ11 = 1 + hx]

(∂xf+ − ∂xf−

1 + hf+

)(x], 0) and γ12 =

1 + hg+(x], 0)

1 + hg−(x], 0).

The stability conditions (SE1) imply x] (∂xf+ − ∂xf−) (x], 0) < 0 and g+(x], 0) < g−(x], 0)

then

1 + hx]

(∂xf+ − ∂xf−

1 + hf+

)(x], 0) < 1, 0 <

1 + hg+(0, 0)

1 + hg−(0, 0)< 1,

for arbitrary h that is to say the fixed point E1 is stable for arbitrary h.

In a same way we obtain that E2 is a stable fixed points for arbitrary h.

Appendix D. Proof of Theorem 6.5

The proof relies on the following classical result:

Lemma Appendix D.1. Roots of the quadratic equation γ2−αγ+β = 0 satisfy |γi| < 1 fori = 1, 2 if and only if the following conditions hold:

(a) 1 + α+ β > 0

(b) 1− α+ β > 0

(c) β < 1

At an equilibrium point (x?, y?) of type E3 the Jacobian matrix is given by

DϕNS,h(x?, y?) =

1 + hx?

(∂xf+−∂xf−

1+hf+

)(x?, y?) hx?

(∂yf+−∂yf−

1+hf+

)(x?, y?)

hy?

(∂xg+−∂xg−

1+hg+

)(x?, y?) 1 + hy?

(∂yg+−∂yg−

1+hg+

)(x?, y?)

(D.1)

18

The trace denoted by TφNS,his given by

TφNS,h= 2 + h

(x?

(∂xf+ − ∂xf−

1 + hf+

)(x?, y?) + y?

(∂yg+ − ∂yg−

1 + hg+

)(x?, y?)

).

and its determinant denoted by DφNS,his equal to

DφNS,h=1 + h

(x?

(∂xf+ − ∂xf−

1 + hf+

)(x?, y?) + y?

(∂yg+ − ∂yg−

1 + hg+

)(x?, y?)

)+ h2

Det (Dϕ(x?, y?))

(1 + hf+)(1 + hg+)(x?, y?)

We verify that1− TφNS,h

+DφNS,h= h2

D

(1 + hf+)(1 + hg+)(x?, y?). (D.2)

The eigenvalues of DϕNS,h(x?, y?) are the roots of the quadratic equation γ2 − TφNS,hγ +

DφNS,h= 0. By Lemma Appendix D.1 we preserve stability if and only if conditions (a), (b)

and (c) are satisfied. We begin with conditions (b) and (c).

Condition (b): We have for all h > 0 that (1 + hf+)(1 + hg+)(x?, y?) > 0 by definitionof f+ and g+. As a consequence, condition (b) is equivalent to D > 0. By assumption, thepoint (x?, y?) is a stable point of (E) so thatD > 0. Condition (b) is then satisfied for all h > 0.

Condition (c): We have

DφNS,h= 1 +

h

(1 + hf+)(1 + hg+)(x?, y?)[T + h(D + C)] , (D.3)

where C is given by

C = x? [(∂xf+ − ∂xf−)g+] (x?, y?) + [y?(∂yg+ − ∂yg−)f+] (x?, y?). (D.4)

The quantity h/[(1 + hf+)(1 + hg+)](x?, y?) is always positive for h > 0. Moreover, asthe equilibrium point (x?, y?) is stable, we have T < 0. As a consequence, the conditionDφNS,h

< 1 is satisfied for arbitrary h > 0 or h sufficiently small depending on the sign ofD+C. Precisely, we must have T +h(D+C) < 0. We have D > 0 by the stability assumptionbut no information on the sign of C. If D + C ≤ 0 the condition is fulfilled for all h > 0 andif D + C > 0 we must have

h < − T

D + C. (D.5)

Condition (a): We have

1 + TφNS,h+DφNS,h

= 4 +2h

[(1 + hf+)(1 + hg+)](x?, y?)[T + h(C +D)] , (D.6)

19

using the previous notations.

By condition (c), we have T +h(C+D) < 0. If all the cases, we must have h sufficiently smallin order to ensure condition (a). Indeed, if T+h(C+D) is strictly negative unconditionally on

h > 0 then condition (a) is satisfied for h < −2[(1 + hf+)(1 + hg+)](x?, y?)

T. If T+h(C+D) <

α < 0 for h < h0 where α does not depend on h then h < −2(1 + hf+)(1 + hg+)](x?, y?)

α.

This concludes the proof.

References

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[2] R. Anguelov and J.M-S. Lubuma. Contributions to the mathematics of the nonstan-dard finite difference method and applications. Numerical Methods Partial DifferentialEquations, 17(5):518–543, 2001.

[3] R. Anguelov, J.M-S. Lubuma, and M. Shillor. Topological dynamic consistency of non-standard finite difference schemes for dynamical systems. Journal of Difference Equationsand Applications, 17(12):1769–1791, 2011.

[4] R. Anguelov, Y. Dumont, J.M-S. Lubuma, and M. Shillor. Dynamically consistent non-standard finite difference schemes for epidemiological models. Journal of Computationaland Applied Mathematics, 255:161–182, 2014.

[5] F. Brauer and C. Castillo-Chavez. Mathematical models in population biology and epi-demiology, volume 1. Springer, 2001.

[6] J. Cartwright and O. Piro. The dynamics of Runge–Kutta methods. International Journalof Bifurcation and Chaos, 2(3):427–449, 1992.

[7] J. Cresson, B. Puig, and S. Sonner. Stochastic models in biology and the invarianceproblem. preprint, 2012.

[8] J. Cresson, B. Puig, and S. Sonner. Validating stochastic models: invariance criteriafor systems of stochastic differential equations and the selection of a stochastic Hodgkin-Huxley type model. International Journal of Biomathematics and Biostatistics, 2:111–122, 2013.

[9] J.P. Demailly. Analyse numérique et équations différentielles. Collection Grenoble sci-ences. EDP Sciences, 2006.

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[10] D.T. Dimitrov and H.V. Kojouharov. Positive and elementary stable nonstandard nu-merical methods with applications to predator-prey models. J. Comput. Appl. Math., 189(1):98–108, 2006.

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22

0

1

2

3

4

5

6

7

-0.2 0 0.2 0.4 0.6 0.8 1

NSFDEULER

RK2RK4

(a) h = 0.01

0

1

2

3

4

5

6

7

-0.2 0 0.2 0.4 0.6 0.8 1

NSFDEULER

RK2RK4

(b) h = 0.1

0

1

2

3

4

5

6

7

-0.2 0 0.2 0.4 0.6 0.8 1

NSFDEULER

RK2RK4

(c) h = 0.2

0

1

2

3

4

5

6

7

-0.2 0 0.2 0.4 0.6 0.8 1

NSFDEULER

RK2RK4

(d) h = 0.3

0

1

2

3

4

5

6

7

-0.2 0 0.2 0.4 0.6 0.8 1

NSFDEULER

RK2RK4

(e) h = 0.4

0

1

2

3

4

5

6

7

-0.2 0 0.2 0.4 0.6 0.8 1

NSFDEULER

RK2RK4

(f) h = 0.5

Figure 2: Numerical simulations of Example 2 with x0 = 0.3, y0 = 7.5.

23

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.4 0.5 0.6 0.7 0.8 0.9 1

NSFDEULER

RK2RK4

(a) h = 0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.4 0.5 0.6 0.7 0.8 0.9 1

NSFDEULER

RK2RK4

(b) h = 0.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.4 0.5 0.6 0.7 0.8 0.9 1

NSFDEULER

RK2RK4

(c) h = 0.3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.4 0.5 0.6 0.7 0.8 0.9 1

NSFDEULER

RK2RK4

(d) h = 0.4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.4 0.5 0.6 0.7 0.8 0.9 1

NSFDEULER

RK2RK4

(e) h = 0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.4 0.5 0.6 0.7 0.8 0.9 1

NSFDEULER

RK2RK4

(f) h = 0.6

Figure 3: Numerical simulations of Example 3 with x0 = 0.4, y0 = 0.4.

24

0.35

0.4

0.45

0.5

0.55

0.6

0 0.1 0.2 0.3 0.4 0.5

NSFDEULER

RK2RK4

(a) h = 0.01

0.35

0.4

0.45

0.5

0.55

0.6

0 0.1 0.2 0.3 0.4 0.5

NSFDEULER

RK2RK4

(b) h = 0.1

0.35

0.4

0.45

0.5

0.55

0.6

0 0.1 0.2 0.3 0.4 0.5

NSFDEULER

RK2RK4

(c) h = 0.5

0.35

0.4

0.45

0.5

0.55

0.6

0 0.1 0.2 0.3 0.4 0.5

NSFDEULER

RK2RK4

(d) h = 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

NSFDEULER

RK2RK4

(e) h = 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

NSFDEULER

RK2RK4

(f) h = 4

Figure 4: Numerical simulations of Example 4 with x0 = 0.4, y0 = 0.4.

25