Electronic Journal of Qualitative Theory of Differential Equations 2004, No. 5, 1-10; http://www.math.u-szeged.hu/ejqtde/

Complete polynomial vector fields in simplexes with application to evolutionary dynamics

by N.M. BEN YOUSIF

ABSTRACT. We describe the complete polynomial vector fields and their fixed points in a finite-dimensional simplex and we apply the results to differential equations of genetical evolution models.

AMS Classification. Primary: 34C07 , secondary: 34C23, 37Cxx.

Key words: polynomial vector field, fixed point, mutation, selection, evolution.

1. Introduction

Well-known genetical models [3],[5],[2] of the time evolution of a closed population consisting of N different species describe the rates r_{1}(t), …, r_{N}(t) of the respective species within the whole population at time t ≥ 0 as the solution of the ordinary system of differential equations dr_{k}(t)/dt = F_{k}(r_{1}(t), …, r_{N}(t)) (k = 1, …, N) where the functions F_{k} are some polynomials of degree at most 3. During a seminar on such models one has launched the question what are the strange consequences of the assumption that the evolution has no starting point in time, in particular what can be stated on the non-changing distributions in that case.

In this paper we provide the complete algebraic description of all polynomial vector fields (with arbitrary degrees) V (x) = (F_{1}(x), …, F_{N}(x)) on IR^{N} which give rise to solutions for the evolution equation defined for all time parameters t ∈ IR and satisfying the natural rate conditions r_{1}(t), …, r_{N}(t) ≥ 0 , ∑_{k=1}^{N} r_{k}(t) = 1 whenever r_{1}(0), …, r_{N}(0) ≥ 0 and ∑_{k=1}^{N} r_{k}(0) = 1 . On the basis of the explicit formulas obtained, we describe the structure or the set of zeros for such vector fields (which correspond to the non-changing distributions).

2. Results

Throughout this work IR^{N} := {(ξ_{1}, …, ξ_{N}) : ξ_{1}, …, ξ_{N} ∈ IR}. denotes the N - dimensional vector space of all real N -tuples. We reserve the notations x_{1}, …, x_{N} for the standard coordinate functions x_{k} : (ξ_{1}, …, ξ_{N}) ↦ ξ_{k} on IR^{N} . Our purpose will be to describe the complete polynomial vector fields on the unit simplex

S := (x_{1} + ⋯ + x_{N} = 1, x_{1}, …, x_{N} ≥ 0)

along with their fixed points for the cases of degree ≤ 3 .

Recall [1] that by a vector field on S we simply mean a function S → IR^{N} . A function φ : S → IR is said to be polynomial if it is the restriction of some polynomial of the linear coordinate functions x_{1}, …, , x_{N} : for some finite system coefficients α_{k_{1}...k_{N}} ∈ IR with k_{1}, …, k_{N} ∈ {0, 1, …}) we can write φ(p) = ∑_{k_{1},...,k_{N}} α_{k_{1}...k_{N}} x_{1}^{k_{1}} ⋯ x_{N}^{k_{N}} (p ∈ S) . In accordance with this terminology, a vector field V on S is a polynomial vector field if its components V_{k} := x_{k} ∘ V (that is V (p) = (V_{1}(p), …, V_{N}(p)) for p ∈ S ) are polynomial functions. It is elementary that given two polynomials P_{m} = P_{m}(x_{1}, …, x_{N}) : IR^{N} → IR (m = 1, 2) , their restrictions to S coincide if and only if the difference P_{1} − P_{2} vanishes on the affine subspace A_{S} := (x_{1} + ⋯ + x_{N} = 1) generated by S . We shall see later (Lemma 3.1) that a polynomial P = P (x_{1}, …, x_{N}) vanishes on the affine subspace M := (γ_{1}x_{1}+⋯+γ_{N}x_{N} = δ) iff P = (γ_{1}x_{1}+⋯+γ_{N}x_{N} −δ)Q(x_{1}, …, x_{N}) for some polynomial Q. Thus polynomial vector fields on S admit several polynomial extensions to IR^{N} but any two such extensions differ only by a vector field of the form (x_{1} + ⋯ + x_{N} − 1)W . A polynomial vector field V : S → IR^{N} is said to be complete in S if for any point p ∈ S there is a (necessarily unique) curve C_{p} : IR → S such that C_{p}(0) = p and \frac{d}{dt}C_{p}(t) = V (C_{p}(t)) (t ∈ IR) .

Our main results are as follows.

2.1. Theorem. A polynomial vector field V : S → IR^{N} is complete in S if and only if with the vector fields

Z_{k} := x_{k} ∑_{j=1}^{N} x_{j}(e_{j} − e_{k}) (k = 1, …, N)

where e_{j} is the standard unit vector e_{j} := (0, …, 0, \overbrace{1}^{j}, 0, …, 0) , we have

V = ∑_{k=1}^{N} P_{k}(x_{1}, …, x_{N})Z_{k}

for some polynomial functions P_{1}, …, P_{N} : IR^{N} → IR .

2.2. Theorem. Given a complete polynomial vector field V of S , there are polynomials δ_{1}, …, δ_{N} : IR^{N−1} → IR of degree less than as that of V such that the vector field

\begin{aligned}Ṽ & := ∑_{k=1}^{N−1} x_{k}[δ_{k}(x_{1}, …, x_{N−1}) − ∑_{ℓ=1}^{N−1} x_{ℓ}δ_{ℓ}(x_{1}, …, x_{N−1})]e_{k}+ \\ & + (x_{1} + ⋯ + x_{N−1} − 1) ∑_{ℓ=1}^{N−1} x_{ℓ}δ_{ℓ}(x_{1}, …, x_{N−1})e_{N} \end{aligned}

coincides with V on S . The points of the zeros of V inside the facial subsimplices S_{K}:=S ∩ (x_{1}, …, x_{K} > 0=x_{K+1} =⋯=x_{N}) (K =1, …, N) can be described as

(∗) \begin{matrix}S_{N} ∩ (V = 0) = S ∩ ⋃_{k=1}^{N−1}(δ_{k}(x_{1}, …, x_{N−1}) = 0), \\ S_{K} ∩ (V = 0) = S_{K} ∩ (δ_{1}(x_{1}, …, x_{N−1}) = ⋯ = δ_{K}(x_{1}, …, x_{N−1}))  (K < N). \end{matrix}

Finally we turn back to our motivativation the genetical time evolution equation for the distribution of species within a closed population. Namely in [2] we have the system

(V)  \begin{matrix}\frac{d}{dt}x_{k} = & (∑_{i=1}^{N} g(i)x_{i} − g(k))x_{k}+ \\ + ∑_{i,j=1}^{N} w(i, j)x_{i}x_{j}[∑_{ℓ=1}^{N} M(i, j, ℓ)ε(i, j, ℓ, k) − x_{k}] \end{matrix}

for describing the behaviour of the rates x_{1}(t), …, x_{N}(t) at time t of the N species of the population. here the terms g(k), M(i, j, ℓ) and ε(i, j, ℓ, k) are non-negative constats with ∑_{ℓ=1}^{N} M(i, j, ℓ) = ∑_{k=1}^{N} ε(i, j, ℓ, k) = 1 . Observe that this can be written as

\frac{d}{dt}x = ∑_{k=1}^{N} g(k)Z_{k} + W

with the vector fields

Z_{k}:=x_{k} ∑_{j=1}^{N}x_{j}(e_{j} −e_{k}), W:= ∑_{i,j,k=1}^{N}w(i, j)x_{i}x_{j}[∑_{ℓ=1}^{N}M(i, j, ℓ)ε(i, j, ℓ, k)−x_{k}]e_{k},

respectively. As a consequence of Theorems 2.1 and 2.2 we obtain the following.

2.3. Theorem. Let N ≥ 3 . Then the time evolution of the population can be retrospected up to any time t ≤ 0 starting with any distribution (x(0), …, x_{N}(0)) ∈ S if and only if the term W vanishes on S , that is if simply d/dt x = ∑_{k=1}^{N} g(k)Z_{k}(x_{1}, …, x_{N}) . In this case the set of the stable distributions has the form

⋃_{γ∈{g(1),...,g(N)}} S ∩ (x_{m} = 0 for m ∉ J_{γ} ) where J_{γ} := {m : g(m) = γ} .

2.4. Corollary. If g(1), …, g(N) ≥ 0 and the vector field (V) is complete in S then

\frac{d}{dt} ∑_{k=1}^{N} g(k)x_{k}(t) ≥ 0

for any solution t ↦ x(t) ∈ S of the evolution equation dx/dt = V (x) .

3. Proof of Theorem 2.1

As in the previous section, we keep fixed the notations e_{1}, …, e_{N}, x_{1}, …, x_{N}, S for the standard unit vectors, coordinate functionals and unit simplex in IR^{N} , and V : IR^{N} → IR^{N} is an arbitrarily fixed polynomial vector field. We write ⟨u, v⟩ := ∑_{k=1}^{N} x_{k}(u)x_{k}(v) for the usual scalar product in IR^{N} .

According to [4, (2,2)], V is complete in S if and only if

V (p) ∈ T_{p}(s) := \{v ∈ IR^{N} : ∃ c : IR → S , c(0) = p , \frac{d}{dt}|_{t=0}c(t) = v\} for all p ∈ S.

By writing

\overline{e} := \frac{1}{N} ∑_{k=1}^{N} e_{k}, u_{k} := e_{k} − \overline{e}, S_{k} := S ∩ (x_{k} = 0) (k = 1, …, N)

for the center, the vectors connecting the vertices with the center and the maximal faces of S , it is elementary that

\begin{matrix}T_{p}(S) = \{v : ⟨v, \overline{e}⟩ = 0\} if p ∈ S \ ⋃_{k=1}^{N} S_{k}), \\ T_{p}(S) = \{v : ⟨v, \overline{e}⟩ = ⟨v, u_{k}⟩ = 0 (k ∈ K_{p})\} if p ∈ ⋃_{k=1}^{N} S_{k} and K_{p} := \{k : p ∈ S_{k}\} \end{matrix}

for any non-empty subset K of {1, …, N}. Since the vector field V is polynomial by assumption, it follows that

V is complete in S ⟺ ⟺ ⟨V (p), \overline{e}⟩ = 0 (p ∈ S) and ⟨V (p), u_{m}⟩ = 0 (p ∈ S_{m}, m = 1, …, N).

Let us write

L_{S} := (x_{1} + ⋯ + x_{N} = 1), L_{S_{m}} := L_{S} ∩ (x_{m} = 0) (m = 1, …, N)

for the hyperplane supporting S , and for the affine submanifolds generated by the faces S_{m} , respectively. Since e_{k} = u_{k} + \overline{e} and since polynomials vanishig on a convex set vanish also on its supporting affine submanifold, equivalently we can say

V is complete in S ⟺ ⟺ ⟨V (p), \overline{e}⟩=0 for p∈L_{S} and ⟨V (p), e_{m}⟩=0 for p∈L_{S_{m}} (m=1, …, N).

If P_{1}, …, P_{N} : IR^{N} → IR are polynomial functions then, with the vector fields Z_{k} := x_{k} ∑_{j=1}^{N} x_{j}(e_{j} − e_{k}) (k = 1, …, N) , we have

⟨∑_{k=1}^{N} P_{k}(p)Z_{k}(p), \overline{e}⟩ = ∑_{k=1}^{N} P_{k}(p)⟨Z_{k}(p), \overline{e}⟩ =

= ∑_{k=1}^{N} P_{k}(p) ⟨Z_{k}(p), \frac{1}{N} ∑_{ℓ=1}^{N} e_{ℓ}⟩ =

= \frac{1}{N} ∑_{k=1}^{N} P_{k}(p) ∑_{j,ℓ=1}^{N} ⟨x_{k}(p)x_{j}(p)(e_{j} − e_{k}), e_{ℓ}⟩ =

= \frac{1}{N} ∑_{k=1}^{N} P_{k}(p)x_{k}(p) ∑_{j: j≠k} ∑_{ℓ=j,k} x_{j}(p)⟨e_{j} − e_{k}, e_{ℓ}⟩ =

= \frac{1}{N} ∑_{k=1}^{N} P_{k}(p)x_{k}(p) ∑_{j: j≠k} x_{j}(p) ∑_{ℓ=j,k}⟨e_{j} − e_{k}, e_{ℓ}⟩ =

= \frac{1}{N} ∑_{k=1}^{N} P_{k}(p)x_{k}(p) ∑_{j: j≠k} x_{j}(p)[1 − 1] = 0

for any point p ∈ IR^{N} (not only for p ∈ S ). On the other hand, if p ∈ S_{m} then x_{m}(p) = 0 and

⟨∑_{k=1}^{N} P_{k}(p)Z_{k}(p), e_{m}⟩ = ∑_{k=1}^{N} P_{k}(p) ⟨Z_{k}(p), e_{m}⟩ =

= ∑_{k=1}^{N} P_{k}(p)x_{k}(p) ∑_{j: j≠k} x_{j}(p)⟨e_{j} − e_{k}, e_{m}⟩ =

= ∑_{k: k≠m} P_{k}(p)x_{k}(p) ∑_{j: j≠k,m} x_{j}(p)⟨e_{j} − e_{k}, e_{m}⟩ = 0.

This means that the vector fields of the form V := ∑_{k=1}^{N} P_{k}Z_{k} with arbitrary polynomials P_{1}, …, P_{N} are complete in S , morover ⟨V (p), \overline{e}⟩ = 0 for all p ∈ IR^{N} .

To prove the remaining part of the theorem, we need the following lemma.

3.1. Lemma. If P : IR^{N} → IR is a polynomial function and 0 ≠ ϕ : IR^{K} → IR is an affine function ^{∗} such that P (q) = 0 for the points q of the hyperplane {q ∈ IR^{N} : ϕ(q) = 0} then ϕ a divisor of P in the sense that P = ϕQ with some (unique) polynomial Q : IR^{N} → IR .

∗ That is ϕ is the sum of a linear functional with a constant.

Proof. Trivially, any two hyperplanes are affine images of each other. In particular there is a one to one affine (i.e linear + constant) mapping A : IR^{N} ↔ IR^{N} such that {q ∈ IR^{N} : ϕ(p) = 0} = A({q ∈ IR^{N} : x_{1}(q) = 0}). Then R := P ∘ A is a polynomial function such that R(q) = 0 for the points of the hyperplane {q ∈ IR^{N} : x_{1}(q) = 0}. We can write R = ∑_{k_{1},...,k_{N} =0}^{d} α_{k_{1},...,k_{N}} x_{1}^{k_{1}} ⋯ x_{N}^{k_{N}} with a suitable finite family of

coefficients α_{k_{1},...,k_{N}} . By the Taylor formula, α_{k_{1},...,k_{N}} = \frac{∂^{k_{1}+···+k_{N}}}{∂x_{1}^{k_{1}} ···∂x_{N}^{k_{N}}} |_{x_{1}=···=x_{N} =0} R. It

follows α_{k_{1},...,k_{N}} = 0 for k_{1} > 0 , since R vanishes for x_{1} = 0 . This means that R = x_{1}R_{0} with the polynomial R_{0} := ∑_{k_{1}=1}^{d} ∑_{k_{2},...,k_{N} =0}^{d} x_{1}^{k_{1}−1}x_{2}^{k_{2}} ⋯ x_{N}^{k_{N}} . By the same argument applied for the polynomial function ϕ of degree d = 1 in place of R, we see that ϕ ∘ A = αx_{1} for some constant (polynomial of degree 0 ) α ≠ 0 . That is ϕ = αx_{1} ∘A^{−1} . Therefore

P = R ∘ A^{−1} = [x_{1}R_{0}] ∘ A^{−1} = (x_{1} ∘ A^{−1})(R_{0} ∘ A^{−1}) = ϕ · (\frac{1}{α}R_{0} ∘ A^{−1}).

Since the inverse of an affine mapping is affine as well, the function Q := \frac{1}{α}R_{0} ∘ A^{−1} is a polynomial which suits the statement of the lemma.

3.2. Corollary. A polynomial vector field Ṽ : IR^{N} → IR^{N} coincides with V on S iff it has the form Ṽ = V + (x_{1} + ⋯ + x_{N} − 1)W with some polynomial vector field W : IR^{N} → IR^{N} .

Proof. Observe that D̃ and V coincide on S iff they coincide on the hyperplane L_{S} supporting S . We can write Ṽ = ∑_{k=1}^{N} P̃_{k}e_{k} resp. V = ∑_{k=1}^{N} P_{k}e_{k} with some scalar valued polynomials P̃_{k} resp. P_{k} and, by the lemma, we have P̃_{k} − P_{k} = 0 on L_{S} iff P̃_{k} − P_{k} = (x_{1} + ⋯ + x_{N} − 1)Q_{k} with some polynomial Q_{k} : IR^{N} → IR (k = 1, …, N) , that is if Ṽ − V = (x_{1} + ⋯ + x_{N} − 1)W with the vector field W := ∑_{k=1}^{N} Q_{k}e_{k} .

Instead of the generic polynomial vector field V complete in S , it is more convenient to study another Ṽ coinciding with V on S but having additional properties. As in the proof of the previous corollary, we decompose V as V = ∑_{k} P_{k}e_{k} . Recall that V (p) ∈ T_{p}(S) ⊂ \{v : ⟨v, \overline{e}⟩ = 0\} for the points p ∈ S . In terms of the component functions P_{k} , this means that \frac{1}{N} ∑_{k=1}^{N} P_{k} = 0 that is P_{N} = − ∑_{k: k≠N} P_{k} on S . On the other hand, x_{1} + ⋯ + x_{N} = 0 that is x_{N} = − ∑_{k: k≠N} on S . Introduce the vector field

Ṽ := ∑_{k=1}^{N} P̃_{k}e_{k}

where

\begin{matrix}P̃_{k} := π_{k}(x_{1}, …, x_{N−1}) := P_{k}(x_{1}, x_{2}, …, x_{N−1}, 1 − x_{1} − ⋯ − x_{N−1}) (k < N), \\ P̃_{N} := π_{N}(x_{1}, …, x_{N−1}) := − ∑_{k=1}^{N−1} P̃_{k} = − ∑_{k=1}^{N−1} π_{k}(x_{1}, …, x_{N−1}). \end{matrix}

By its construction, Ṽ coincides with V on S , it is a polynomial of the same degree as V but only in the variables x_{1}, …, x_{N−1} and it has the property ∑_{k=1}^{N} P̃_{k} = 0 on the whole space IR^{N} . The relations Ṽ(p) = V (p) ∈ T_{p}(S) ⊂ {v : ⟨v, e_{k}⟩ = 0} for p ∈ S_{k} (k = 1, …, N) mean

P̃_{k}(p) = ⟨Ṽ(p), e_{k}⟩ = 0 for p ∈ S_{k} = (x_{k} = 0, x_{1} + ⋯ + x_{N} = 1, x_{1}, …, x_{N} ≥ 0).

In terms of the polynomials π_{k} of N − 1 variables this can be stated as

(∗∗) π_{k}(ξ_{1}, …, ξ_{N−1}) = 0 whenever ξ_{k} = 0 (k = 1, …, N − 1) and

− ∑_{k=1}^{N−1} π_{k}(ξ_{1}, …, ξ_{N−1})[= π_{N}(ξ_{1}, …, ξ_{N−1})] = 0 whenever ξ_{1} + ⋯ + ξ_{N−1} = 1.

By the lemma (applied with N −1 instead of N ), the first N −1 equations are equivalent to

π_{k}(ξ_{1}, …, ξ_{N−1}) = ξ_{k}ϱ_{k}(ξ_{1}, …, ξ_{N−1}) (k = 1, …, N − 1)

with some polynomials ϱ_{k} : IR^{N−1} → IR with degree less than the degree of π_{k} and V . Also by the lemma (with N − 1 instead of N ), the last equation can be interpreted as

− ∑_{k=1}^{N−1} π_{k}(ξ_{1}, …, ξ_{N−1}) = π_{N}(ξ_{1}, …, ξ_{N−1}) = [1 − (ξ_{1} + ⋯ + ξ_{N−1})]ϱ_{N}(ξ_{1}, …, ξ_{N−1})

with some polynomial ϱ_{N} : IR^{N−1} → IR of degree less than that of V . Thus

\begin{matrix}− ∑_{k=1}^{N−1} ξ_{k}ϱ_{k}(ξ_{1}, …, ξ_{N−1}) = [1 − (ξ_{1} + ⋯ + ξ_{N−1})]ϱ_{N}(ξ_{1}, …, ξ_{N−1}), \\ ∑_{k=1}^{N−1} ξ_{k}[ϱ_{N} − ϱ_{k}](ξ_{1}, …, ξ_{N−1}) = ϱ_{N}(ξ_{1}, …, ξ_{N−1}). \end{matrix}

By introducing the polynomials δ_{k} := ϱ_{k} − ϱ_{N} (k = 1, …, N − 1) of N − 1 variables, we can reformulate the relationships (∗∗) as

π_{k} = ξ_{k}ϱ_{k} = ξ_{k}(δ_{k} + ϱ_{N}) (k = 1, …, N − 1),
π_{N} = (1 − ξ_{1} − ⋯ − ξ_{N})ϱ_{N},
ϱ_{N} = −ξ_{1}δ_{1} − ⋯ − ξ_{N−1}δ_{N−1}

which is the same as

(∗ ∗ ∗) \begin{matrix}π_{k}(ξ_{1}, …, ξ_{N−1}) = ξ_{k}[δ_{k}(ξ_{1}, …, ξ_{N−1}) − ∑_{ℓ=1}^{N−1} ξ_{ℓ}δ_{ℓ}(ξ_{1}, …, ξ_{N−1})] for k ≠ N, \\ π_{N} = (ξ_{1} + ⋯ + ξ_{N} − 1) ∑_{ℓ=1}^{N−1} ξ_{ℓ}δ_{ℓ}(ξ_{1}, …, ξ_{N−1}) \end{matrix}

where δ_{1}, …, δ_{N−1} are arbitrary polynomials of the variables ξ_{1}, …, ξ_{N−1} .

Summarizing the arguments, we have obtained the followig result.

3.3. Proposition. Let V = ∑_{k=1}^{N} P_{k}e_{k} be a vector field where P_{1}, …, P_{N} : IR^{N} → IR are polynomials of the coordinate functions x_{1}, …, x_{N} . Then V is complete in the simplex S := (x_{1} + ⋯ + x_{N} = 1, x_{1}, …, x_{N} ≥ 0) if and only if there exist polynomials δ_{1}, …, δ_{N−1} of N − 1 variables and degree less than that of V such that the vector field Ṽ := ∑_{k=1}^{N} π_{k}(x_{1}, …, x_{N−1})e_{k} , where the polynomials π_{k} are given by (∗ ∗ ∗) in terms of δ_{1}, …, δ_{N−1} , coincides with V on the hyperplane L_{S} := (x_{1} + ⋯ + x_{N} = 1) .

On the basis of the proposition we can finish the proof of Theorem 2.1 as follows. Let V be a polynomial vector field complete in S . By the proposition, we can find a vector field Ṽ of the form (*) coinciding with V on S such that where δ_{1}, …, δ_{N−1} : IR^{N−1} → IR are polynomials. It suffices to show that the vector field

V̂ := − ∑_{k=1}^{N−1} δ(x_{1}, …, x_{N−1})Z_{k}(x_{1}, …, x_{N}) = ∑_{k=1}^{N−1} δ(x_{1}, …, x_{N−1}) ∑_{ℓ=1}^{N} x_{k}x_{ℓ}(e_{k} − e_{ℓ})

coincides with Ṽ on S . Consider any point p ∈ S and let ξ_{k} := x_{k}(p) (k = 1, …, N) . Since ξ_{N} = 1 − ξ_{1} − ⋯ − ξ_{N−1} , it is straightforward to check that indeed

Ṽ(p) − V̂(p) = ∑_{k=1}^{N−1} ξ_{k}[δ_{k}(ξ_{1}, …, ξ_{N−1}) − ∑_{ℓ=1}^{N−1} ξ_{ℓ}δ_{ℓ}(ξ_{1}, …, ξ_{N−1})]e_{k}+

+ (ξ_{1} + ⋯ + ξ_{N−1} − 1) ∑_{ℓ=1}^{N−1} ξ_{ℓ}δ_{ℓ}(ξ_{1}, …, ξ_{N−1})e_{N}+

+ ∑_{k=1}^{N−1} δ(ξ_{1}, …, ξ_{N−1}) ∑_{ℓ=1}^{N} ξ_{k}ξ_{ℓ}(e_{k} − e_{ℓ}) = 0 .

4. Proof of Theorem 2.2

According to Proposition 3.3, we can take a vector field Ṽ of the form (∗) coinciding with V on S where δ_{1}, …, δ_{N} : IR^{N−1} → IR are polynomials of degree less than that of V . Consider a point p := (ξ_{1}, …, ξ_{N}) ∈ S . Necessarily ξ_{N} = 1 − ξ_{1} − ⋯ − ξ_{N−1} ≥ 0 and ξ_{1}, …, ξ_{N−1} ≥ 0 . We have V (p) = 0 iff

ξ_{k}[δ_{k}(ξ_{1}, …, ξ_{N−1}) − ∑_{ℓ=1}^{N−1} ξ_{ℓ}δ_{ℓ}(ξ_{1}, …, ξ_{N−1})] = 0 (k = 1, …, N − 1).

Assume these equations hold with ξ_{1}, …, ξ_{N} >0 , that is p∈S_{N} . Then δ_{1}(ξ_{1}, …, ξ_{N−1})= ⋯ = δ_{N−1}(ξ_{1}, …, ξ_{N−1}) = ∑_{ℓ=1}^{N−1} ξ_{ℓ}δ_{ℓ}(ξ_{1}, …, ξ_{N−1}) . However, by writing δ for the common value of the δ_{k}(ξ_{1}, …, ξ_{N−1}) , we have δ =∑_{ℓ=1}^{N−1}ξ_{ℓ}δ, that is ξ_{N}δ =(1−∑_{ℓ=1}^{N−1}ξ_{ℓ})δ =0 and δ = 0 .

Assume finally that K < N and ξ_{1}, …, ξ_{K} > 0 = ξ_{K+1} = ⋯ = ξ_{N} , that is p ∈ S_{K} . Then V (p) = 0 iff

δ_{k}(ξ_{1}, …, ξ_{K}, 0, …, 0) = ∑_{ℓ=1}^{K} ξ_{ℓ}δ_{ℓ}(ξ_{1}, …, ξ_{K}, 0, …, 0) (k = 1, …, K).

Again the δ_{k}(ξ_{1}, …, ξ_{K}, 0, …, 0) assume a common value δ. However, in this case ∑_{ℓ=1}^{K} ξ_{ℓ} = 1 and hence δ may be arbitrary for V (p) = 0 .

5. Proof of Theorem 2.3

First we check that ∑_{k=1}^{N}(∑_{i=1}^{N} g(i)x_{i} − g(k))x_{k}e_{k} = ∑_{k=1}^{N} g(k)Z_{k} on S . Indeed, given any index m, from the fact that ∑_{i=1}^{N} x_{i} = 1 on S , it follows

⟨∑_{k=1}^{N} g(k)Z_{k} , e_{m}⟩ = ∑_{k=1}^{N} g(k)⟨Z_{k}, e_{m}⟩ =

= ∑_{k=1}^{N} g(k)⟨x_{k} ∑_{i=1}^{N} x_{i}(e_{i} − e_{k}), e_{m}⟩ = ∑_{i,k=1}^{N} g(k)x_{k}x_{i}⟨e_{i}, e_{m}⟩ − ∑_{i,k=1}^{N} g(k)x_{k}x_{i}⟨e_{k}, e_{m}⟩ =

= ∑_{k=1}^{N} g(k)x_{k}x_{m} − g(m)x_{m} ∑_{i=1}^{N} x_{i} =

= (∑_{i=1}^{N} g(i)x_{i} − g(m))x_{m} = ⟨∑_{k=1}^{N}(∑_{i=1}^{N} g(i)x_{i} − g(k))x_{k}e_{k} , e_{m}⟩ .

Since, in general, (real-)linear combinations of complete vector fields are complete vector fields (see e.g. [1]), and since the Z_{k} are complete in S , the field V = ∑_{k=1}^{N} g(k)Z_{k} − W is complete in S iff W is complete in S . As we have seen, the polynomial vector field W is complete in S iff ⟨W, ∑_{k=1}^{N} e_{k}⟩ = 0 and ⟨W (x_{1}, …, x_{N}), e_{k}⟩ = 0 whenever x_{k} = 0 for some index k and ∑_{i=1}^{N} x_{i} = 1 . It is well known that, by its construction, ⟨V (x_{1}, …, x_{N}), ∑_{i=1}^{N} e_{i}⟩ = 0 and hence ⟨W (x_{1}, …, x_{N}), ∑_{i=1}^{N} e_{i}⟩ = 0 if ∑_{i=1}^{N} x_{i} even in the case if V is not complete in S . Fix any index k. By the definition W := ∑_{m=1}^{N} ∑_{i,j=1}^{N} w(i, j)x_{i}x_{j}[∑_{ℓ=1}^{N} M(i, j, ℓ)ε(i, j, ℓ, m) − x_{m}]e_{m} we have ⟨W (p), e_{k}⟩ = 0 for all points p := (ξ_{1}, …, ξ_{N}) with ξ_{k} = 0 and ∑_{i=1}^{N} ξ_{i} = 1 if and only if

∑_{{i,j=1 \atop i,j≠k}}^{N} w(i, j)ξ_{i}ξ_{j} ∑_{ℓ=1}^{N} M(i, j, ℓ)ε(i, j, ℓ, k) = 0 if ∑N_{{i=1 \atop i≠k}} ξ_{i} = 1 .

By elementary properties of bilinear forms, this latter relation holds iff

w(i, j) ∑_{ℓ=1}^{N} M(i, j, ℓ)ε(i, j, ℓ, k) = 0 if i, j ≠ k .

Since N ≥ 3 , the field W has this property for all indices k = 1, …, N iff all these terms vanish and hence W = 0 . Thus V is complete in S iff W = 0 that is V = ∑_{k=1}^{N} g(k)Z_{k} on S . In this case, the equation V (ξ_{1}, …, ξ_{N}) = 0 with (ξ_{1}, …, ξ_{N}) ∈ S means

ξ_{k}(∑_{i=1}^{N} g(i)ξ_{i} − g(k)) = 0 (k = 1, …, N)

along with the conditions ξ_{1} + ⋯ + ξ_{N} = 1 and ξ_{1}, …, ξ_{N} ≥ 0 . Consider a point (ξ_{1}, …, ξ_{N}) ∈ S and write J := {j : ξ_{j} > 0}. Then ξ_{k} = 0 for k ∉ J and hence ∑_{j∈J} ξ_{j} = 1 and V (ξ_{1}, …, ξ_{N}) = 0 iff g(k) = ∑_{i=1}^{N} ξ_{i}g(i) = ∑_{j∈J} ξ_{j}g(j) for the indices k ∈ J . By writing γ := ∑_{j∈J} ξ_{j}g(j) for the common value of the g(k) with (k ∈ J) , we see that V (ξ_{1}, …, ξ_{N}) = 0 for any ξ_{1}, …, ξ_{N}) ∈ S ∩ ⋂_{j∈J}(x_{j} > 0) ∩ ⋂_{i∉J}(x_{i} = 0) . This completes the proof.

REFERENCES

[1] F. Brickell - R.S. Clark, Differentiable Manifolds, Van Nostrand Reinhold Co., London, 1970.

[2] L. Hatvani, A modification of Tusnády’s modell for genenetical evolution, preprint, 2001.

[3] J. Hofbauer – K. Sigmund, Evolutionary Games and Replicator Dynamics, Cambridge Univ. Press, Cambridge, 1998.

[4] L.L. Stachó, On nonlinear projections of vector fields, NLA98: Convex analysis and chaos (Sakado, 1998), 47–54, Josai Math. Monogr., 1, Josai Univ., Sakado, 1999.

[5] G. Tusnády, Mutation and selection (Hungarian), Magyar Tudomány, 7 (1997), 792-805.

Address: Department of Mathematics, Alfatex University, Tripoli, Libya Email: nuri mofideh@math.u-szeged.hu