On a Geometrical Study of Population Ecosystems

Mihai Postolache and Oana Mocioalcă

Abstract We study a class of Lyapunov functions to be applied for investigation of the stability of population ecosystems. In a two dimensional case, a locally geometrical study is given.

Mathematics Subject Classification: 92D25, 94A17 Key words: steady-state, globally stable, extinction.

1 Introduction

In this paper, we shall survey some contribution to the geometrical study of mathematical models of ecosystems with emphasis upon some classes of models of interacting species. Initially investigated by A.J. Lotka [?] and V. Volterra [?], this class of ecosystems plays an important role in mathematical ecology. The models considered in this paper are of the form

ẋ_{i} = x_{i}F_{i}(x_{1}, x_{2}, …, x_{n}), i = 1, 2, …, n, (1)

where x_{i} is the density of the i^{th} species in the community at time t, and ẋ_{i} denotes \frac{dx_{i}}{dt}. Each F_{i} is a continuous function from R_{+}^{n}, the nonnegative cone in R^{n}, to R and is sufficiently smooth to guarantee that initial value problems associated with (??) have unique solutions in the population orthant, R_{+}^{n}.

As a rule, the stability of such a system is studied by using the eigenvalues of its linear approximation. This method gives answer only concerning the stability relative to infinitesimal perturbations of the initial state. In real cases these systems are subjected to large perturbations. So, the study of stability relative to finite perturbations is useful. This requires an extension from a local property to a global concept. We shall denote by x^{∗} = (x_{1}^{∗}, x_{2}^{∗}, …, x_{n}^{∗}) a steady-state of (??). x^{∗} is said to be globally (asymptotically) stable if for any neighbourhood U of x^{∗}, there exists a neighbourhood W of x^{∗} such that any orbit through W remains forever in U and x(t) → x^{∗} for all x ∈ W as the time variable tends to infinity.

Balkan Journal of Geometry and Its Applications, Vol.2, No.2, 1997, pp. 101-108 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

In this period, much effort in studies of ecological models has been directed to local stability analysis and numerical developments [?]. To investigate a global phenomenon such as globally asymptotically stability analytical studies are required.

Based on some ideas of Hsu [?], and using the direct method of Lyapunov [?], we shall study a class of Lyapunov functions to be applied for the study of globally asymptotically stability of systems of the form (??). In a two dimensional case, a locally geometrical study of (??) is given. We prove that, in general, totally extinction cannot appear in (??) and introduce a Riemannian structure to study it.

2 An extension of the direct method of Lyapunov

We have [?], the following

Definition 2.1 For (??), a continuous function V: R^{n} → R is a Lyapunov function if

1) V is positive definite with V (x^{∗}) = 0;

2) lim_{x_{i}→0^{+}} V (x) = ∞; lim_{x_{i}→∞} V (x) = ∞ for each i, i = 1, 2, …, n;

3) V̇ (x) = ∑_{i=1}^{n} (\frac{∂V}{∂x_{i}} ) x_{i}F_{i}(x) is nonpositive for all positive values of x.

Remark 2.1 The third condition in Definition ?? shows that the time derivative of V (x) along every solution of (??) is nonpositive.

From LaSalle’s extension of the direct method of Lyapunov [?], we have the following

Theorem 2.1 The positive steady-state x^{∗} of (??) is globally asymptotically stable if there exists a Lyapunov function V (x) such that V̇ (x) is nonpositive in the positive orthant and it does not vanish identically along a nontrivial solution of (??) except for the constant solution x(t) = x^{∗}.

Let V_{i}(x_{i}) be the Lyapunov function for a stable single species model whose population is x_{i}. If (??) describes a multispecies system in which each species is self-regulating and the interspecific interactions are relatively weak, then the Lyapunov function can be of the form

V (x) = ∑_{i=1}^{n} c_{i}V_{i}(x_{i}), (2)

where c_{1}, c_{2}, …, c_{n} are positive constants. These constants are chosen so that V̇ (x) is nonpositive in the positive orthant. Hsu [?], suggested that a general expression for V_{i}(x_{i}) is of the form

V_{i}(x_{i}) = ∫_{x_{i}^{∗}}^{x_{i}} \frac{h_{i}(s)}{g_{i}(s)}ds (3)

where h_{i} and g_{i} have the following properties:

a) h_{i} and g_{i} are continuous functions such that for every i, i = 1, 2, …, n we have h_{i}(s) < 0 for all s ∈ (0, x_{i}^{∗}), h_{i}(x_{i}^{∗}) = 0, h_{i}(s) > 0 for all s ∈ (x_{i}^{∗}, ∞);

b) g_{i}(s) > 0 for all s > 0.

c) h_{i} and g_{i} are chosen such that

lim_{x_{i}→0^{+}} V (x) = ∞; lim_{x_{i}→∞} V (x) = ∞, i = 1, 2, …, n.

Taking into account (??) and (??) we obtain

V (x) = ∑_{i=1}^{n} c_{i} ∫_{x_{i}^{∗}}^{x_{i}} \frac{h_{i}(s)}{g_{i}(s)}ds

where h_{i} and g_{i} satisfy a), b) and c).

Remark 2.2 The simplest expressions for the functions h_{i} and g_{i} are h_{i}(s) = s − x_{i}^{∗} and g_{i}(s) = s.

Using these functions and based on Theorem ?? we can state

Theorem 2.2 A positive steady-state of (??) at x^{∗} is globally asymptotically stable if there exist positive constants c_{1}, c_{2}, …, c_{n} such that the function

V̇ (x) = ∑_{i=1}^{n} c_{i}(x_{i} − x_{i}^{∗})F_{i}(x)

is negative semidefinite in the positive orthant and V̇ does not vanish identically along a nontrivial solution of (??) except for the constant solution x(t) = x^{∗}.

Remark 2.3 It can be shown that for a general model of a population

ẋ = xF (x), (4)

where F: R_{+} → R is a continuous function, a Lyapunov function is

V (x) = x − x^{∗} − ln \frac{x}{x^{∗}} ,

and V̇ (x) = (x − x^{∗})F (x).

So, the positive steady-state x^{∗} of (??) is globally asymptotically stable if F (x) > 0 for all x ∈ (0, x^{∗}) and F (x) < 0 for all x ∈ (x^{∗}, ∞).

Let us consider now a community of n interacting species modelled by a Lotka-Volterra system. This system is of the form (??) where

F_{i}(x_{1}, x_{2}, …, x_{n}) = b_{i} + ∑_{k=1}^{n} a_{ik}x_{k}, i = 1, 2, …, n. (5)

Here b_{i}, −a_{ii} are positive constants and a_{ik}, i ≠ k are constants with any sign. As we shall see later, any arbitrary sign for a_{ik}, i ≠ k, allows us a greater flexibility for the interactions between the i^{th} and k^{th} species in the community. If we define A = (a_{ik}) and b = (b_{1}, b_{2}, …, b_{n}) then it can be shown that x^{∗} = −A^{−1}b^{t} is a steady-state of system. Let us suppose that x^{∗} ∈ R_{+}^{n} is positive and C = diag(c_{1}, c_{2}, …, c_{n}). Using Theorem ?? we can investigate the stability of the steady-state x^{∗} > 0 of this system. By Remark ?? we deduce that

V (x) = ∑_{i=1}^{n} c_{i}(x_{i} − x_{i}^{∗} − x_{i}^{∗} ln \frac{x_{i}}{x_{i}^{∗}})

can be used as a Lyapunov function. Clearly, V (x) satisfies the conditions V (x^{∗}) = 0, V (x) > 0 for all x ∈ R_{+}^{n}, x ≠ x^{∗}, V (x) → ∞ as x → ∞ and x → 0. We have

V̇ (x) = ∑_{i=1}^{n} c_{i}(x_{i} − x_{i}^{∗})(b_{i} + ∑_{k=1}^{n} a_{ik}x_{k})

= ∑_{i=1}^{n} c_{i}(x_{i} − x_{i}^{∗})(∑_{k=1}^{n} a_{ik}(x_{k} − x_{k}^{∗}))

= \frac{1}{2}(x − x^{∗})^{t}(CA + A^{t}C)(x − x^{∗}).

Let the matrix CA + A^{t}C be negative semidefinite. Then, it is clear that x = x^{∗} is asymptotically stable. In fact, the basin of atraction of x^{∗} is R_{+}^{n}. Therefore we have outlined a proof of this result

Theorem 2.3 The steady-state x^{∗} is globally assymptotically stable if there exists a positive diagonal matrix C such that CA + A^{t}C is negative semidefinite and the function

V̇ (x) = \frac{1}{2}(x − x^{∗})^{t}(CA + A^{t}C)(x − x^{∗})

does not vanish identically along a nontrivial solution.

3 Stability via Poincaré transformation

Let V̇ be the function in Theorem ??. We remark that is desirable for \frac{∂F_{1}}{∂x_{1}} (x^{∗}) to be nonpositive. If it is positive, Hsu recomments to subject (1.1) to the following Poincaré transformation [?],

x_{1} = \frac{1}{y_{1}} , x_{k} = \frac{y_{k}}{y_{1}} , k = 2, 3, …, n. (6)

We obtain

ẏ_{1} = −y_{1}F_{1}(\frac{1}{y_{1}} , \frac{y_{2}}{y_{1}} , …, \frac{y_{n}}{y_{1}} )

(7)

ẏ_{k} = y_{k}(F_{k} − F_{1}), k = 2, 3, …, n.

Remark 3.1 The transformation (??) maps x^{∗} into y^{∗} as follows

y^{∗} = (\frac{1}{x_{1}^{∗}} , \frac{x_{2}^{∗}}{x_{1}^{∗}} , …, \frac{x_{n}^{∗}}{x_{1}^{∗}}).

Now, if we apply Theorem ?? we find

V̇ (y) = −c_{1}(y_{1} − y_{1}^{∗})F_{1} + ∑_{k=2}^{n} c_{k}(y_{k} − y_{k}^{∗})(F_{k} − F_{1}). (8)

In this context, we get

Theorem 3.1 The steady-state y^{∗} is globally asymptotically stable if there exists positive constants c_{1}, c_{2}, …, c_{n} such that the function (??) is negative semidefinite in the positive orthant and V̇ (y) does not vanish identically along of nontrivial solution of (??).

We wish now to consider the system [?],

ẋ_{1} = X(x_{1}, x_{2}); ẋ_{2} = Y (x_{1}, x_{2}) (9)

where

X(x_{1}, x_{2}) = −2(α_{2} + α_{1}L)x_{1}x_{2} − (Lα_{2} − α_{1})x_{2}^{2} − (α_{1} − Lα_{2})x_{1}^{2} + λx_{1}
Y (x_{1}, x_{2}) = −2(α_{1} − α_{2}L)x_{1}x_{2} + (α_{2} + Lα_{1})x_{1}^{2} − (α_{2} + Lα_{1})x_{2}^{2} + λx_{2}.

This system describes a 2-coral community model of scleractinian corals. It can be shown that

x_{1}^{∗} = \frac{λ(α_{1} − α_{2}L)}{(α_{1} − α_{2}L)^{2} + (α_{2} + α_{1}L)^{2}} ; x_{2}^{∗} = \frac{α_{2} + α_{1}L}{α_{1} − α_{2}L}x_{1}^{∗} (10)

is a positive steady-state of (??). Moreover, by Theorem ?? and a theorem of Antonelli and Lin [?], we get

Theorem 3.2 The system (??) exhibits a positive steady-state at (x_{1}^{∗}, x_{2}^{∗}) given by (??). Moreover, this steady-state is unique and globally asymptotically stable for the set R_{+}^{2}.

4 A locally geometrical study

According to [?], extinction occurs in (??) if there is a solution (x_{1}, x_{2}, …, x_{n}) of (??) with x_{i}(0) > 0, i = 1, …, n and having a component x_{j} which satisfies lim_{t→τ} x_{j}(t) = 0 for some τ in (0, +∞).

Let us consider a community of n ≥ 2 mutually competing species that is a system (??) whose function F in (??) has the coefficients such that a_{ik}, a_{ki}, i ≠ k are both negative, that is

ẋ_{i} = x_{i}(b_{i} − ∑_{j=1}^{n} a_{ij}x_{j}), i = 1, …, n. (11)

Consider also the auxiliary function ρ(t) = x_{1}(t)x_{2}(t) ⋯ x_{n}(t) and its total time derivative along trajectories of (??). We have

ρ̇(t) = ρ [∑ b_{i} − (∑ a_{i1})x_{1} − (∑ a_{i2})x_{2} − ⋯ − (∑ a_{in}) x_{n}]

Let us suppose that total extinction occurs. Then there exists a T such that whenever t ≥ T

x_{1}(t) < \frac{∑ b_{i}}{(n + 1) ∑ a_{i1}} ; …; x_{n} < \frac{∑ b_{i}}{(n + 1) ∑ a_{in}} .

It follows that for η = \frac{1}{n+1} ∑ b_{i} we have ρ̇ ≥ ρη, therefore ρ(t) ≥ ρ(T)e^{η(t−T)}, t ≥ T. This implies that lim_{t→∞} ρ(t) = ∞. We get the following result which states that complete extinction cannot occur in (??).

Theorem 4.1 For any solution (x_{1}(t), …, x_{n}(t)) of a mutually competing system with x_{1}(0) > 0, …, x_{n}(0) > 0 the limits lim_{t→∞} x_{i}(t) = 0, i = \overline{1, n} cannot occur simultaneously.

In the following we consider n = 2 and we would like to point out how to use the local Riemannian geometrical structure and information geometry for studying the qualitative fluctuations of our system. We have

Proposition 4.1 The two dimensional mutually competing system is Hamiltonian if and only if the following conditions hold

b_{1} + b_{2} = 0, a_{21} + 2a_{11} = 0, a_{12} + 2a_{22} = 0.

In this case we find the Hamiltonian

H(x_{1}, x_{2}) = x_{1}x_{2}(b_{1} − a_{11}x_{1} + a_{22}x_{2}). (12)

We easily check that H(x_{1}, x_{2}) verifies the Hamilton equations

\frac{∂H}{∂x_{2}} = \frac{dx_{1}}{dt} , −\frac{∂H}{∂x_{1}} = \frac{dx_{2}}{dt} .

We shall denote by (x_{1}^{∗}, x_{2}^{∗}) the nontrivial steady-states of (??). For our purpose we make the expansion around (x_{1}^{∗}, x_{2}^{∗}) of H given by (??) according to Taylor formula to the second order. We obtain

−a_{11}x_{2}^{∗}(x_{1}−x_{1}^{∗})+a_{22}x_{1}^{∗}(x_{2}−x_{2}^{∗})+(b_{1}−2a_{11}x_{1}^{∗}+2a_{22}x_{2}^{∗})(x_{1}−x_{1}^{∗})(x_{2}−x_{2}^{∗}) = H−H_{0}.

We now introduce the Boltzman-Gibbs entropy, in fact the absolute uncertainty of a state P

S(P) = − ∫_{Ω} P ln P dσ,

where Ω is the state space, P is the density function of a random vector (X, Y), and S(P) is considered as an absolute information (entropy). Also, we introduce the Kullbak entropy or the relative information

S(Q|P) = ∫_{Ω} P (ln P − ln Q)dσ. (13)

Expression (??) is called the relative uncertainty of P with respect to Q and represents an information distance between P and Q. Let us consider a two dimensional differentiable manifold M of class C^{∞} with local coordinates u : (u^{1} = X, u^{2} = Y) and local metric defined by S(Q|P) of C^{∞} class. Thus P (X, Y) → P (u), Q(X, Y) → Q(u), and they are functions of C^{∞} class. The form (??) may be transformed into the following expression

S(u, v) = S(P (v)|P (u)) = ∫_{M} P (u)(ln P (u) − ln P (v))dω. (14)

If we take into account that

∫_{M} P (u)dω = ∫_{M} P (v)dω = 1

we obtain

∫_{M} \frac{∂P}{∂u^{α}} dω = 0, ∫_{M} \frac{∂^{2}P}{∂u^{α}∂u^{β}} dω = 0, α, β ∈ \{1, 2\}.

Then the first and the second derivatives of (??) with respect to u are found to be given by

\frac{∂S}{∂u^{i}} |_{u=v} = 0, \frac{∂^{2}S}{∂u^{i}∂u^{k}} |_{u=v} =< \frac{∂ ln P}{∂u^{i}} \frac{∂ ln P}{∂u^{k}} >= 2g_{ik} (15)

We remark that (??) is positive definite, therefore the symmetric matrix g = (g_{ik}) is positive definite. In particular, det(g) > 0 or rank(g) = 2. Thus we may locally write ds^{2} = g_{αβ}du^{α}du^{β} ≥ 0, where ds is the information distance between the states parametrized by u and u + du in the two dimensional state manifold M. Therefore, M becomes locally a Riemannian manifold.

We introduce now the Gibbs distribution [?]

P (x_{1}, x_{2}) = Z^{−1}(x_{1}, x_{2}) exp(−H(x_{1}, x_{2}) + H(x_{1}^{∗}, x_{2}^{∗})), (16)

where Z(x_{1}, x_{2}) can be written as

Z(x_{1}, x_{2}) = ∫∫_{−∞}^{+∞} exp(−H(x_{1}, x_{2}) + H(x_{1}^{∗}, x_{2}^{∗}))dx_{1} dx_{2}.

If we suppose that b_{1} + a_{11}a_{22}x_{1}^{∗}x_{2}^{∗} + 2a_{22}x_{2}^{∗} − 2a_{11}x_{1}^{∗} > 0 then

Z = \frac{π}{\sqrt{b_{1} + a_{11}a_{22}x_{1}^{∗}x_{2}^{∗} + 2a_{22}x_{2}^{∗} − 2a_{11}x_{1}^{∗}}} .

If we use the expression (??) in (??) then we can compute the metric tensor and Gauss curvature.

Acknowledgements. A version of this paper was presented at the First Conference of Balkan Society of Geometers, Politehnica University of Bucharest, September 23-27, 1996.

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Politehnica University of Bucharest Department of Mathematics I Splaiul Independenţei 313 77206 Bucharest, Romania e-mail: postol@math.pub.ro