Global stability of a periodic
Holling-Tanner predator-prey model
Benedetta Lisena
Dipartimento di Matematica, Universit´a degli studi di Bari, 70125 Bari, ITALY
Abstract.
This paper is concerned with the global dynamics of a Holling-Tanner predator-prey model with periodic coefficients. We establish sufficient conditions for the existence of a positive solution and its global asymptotic stability. The stability conditions are first given in average form and afterward as pointwise estimates. In the autonomous case, the previous criteria lead to a known result.
Keywords: Predator-prey; Global stability; Periodic solution; Holling-type II.
1
Introduction
In population dynamics to investigate the relationship between two biological species of predator-prey type, many mathematical models have been introduced. Leslie proposed the following model
(x0 = x(a − b x) − p(x) y
y0= y(c − dy x) ,
(1.1)
where x, y stand for the population of the prey and the predator, respectively, p(x) is called the predator functional response to prey and dy
x is known as the Leslie-Gower term. The functional response
p(x) = k x x + m
is known as functional response of Holling type II; the parameter k is the maxi-mal predator per capita consumption rate and m is the number of prey necessary to achieve one-half of the maximum rate k. The corresponding model
x0= x(a − b x) − k xy x + m y0 = y(c − dy x) (1.2)
is also called Holling-Tanner predator-prey model. In (1.2), the prey has logistic growth with carrying capacity ab in absence of predator, while the predator
equation follows a logistic growth curve with dynamic carrying capacity c xd , since it is proportional to the prey population size. For further information on models (1.1) and (1.2) from the viewpoint of ecology see [6, 10, 11]. Hsu and Huang[6] study the asymptotic stability of the positive equilibrium of system (1.2) and investigate its global stability by applying the Dulac’s criterion and the Lyapunov method. The second equation of (1.2) presents a certain singularity at x = 0. To avoid such a singularity, in [2], the authors propose a modified Holling-Tanner model. This model has been widely studied by several researchers, for instance [4, 12, 13]. In differential systems (1.1) and (1.2) all biological and environmental parameters are constant in time. However such parameters are naturally subject to fluctuation in time that can be periodic. In order to take into account such fluctuations, a model must be nonautonomous and its study requires more refined tools. In this paper, we consider the periodic Holling-Tanner system u0= u(a(t) − u − v u + m(t)) v0= v(b(t) − v γ(t) u) , (1.3)
in which u(t), v(t) represent the population of the prey and the predator respec-tively, the time t appears explicitly in the biological parameters a(t), b(t), m(t) and γ(t).
In [10] the authors study the global dynamics of the following predator-prey system
(x0= x(a(t) − b(t) x) − c(t, x) y
y0= y(d(t) − e(t)y x) ,
(1.4)
including (1.3). In particular, in the periodic case, they establish sufficient criteria for the global asymptotic stabiliy of the periodic solution (x∗(t), y∗(t)). Their approach is developed by the Lyapunov function
V (t) = | ln(x(t)) − ln(x∗(t))| + | ln(y(t)) − ln(y∗(t))| , (1.5) which is common for nonautonomous population models of various types. Predator-prey systems of periodic nature have often been studied beginning from Lotka-Volterra systems. In [8], Lisena tackles the question of global attractivity for the following periodic model
(
x0= x(a1(t) − b11(t) x − b12(t) y)
y0 = y(−a2(t) + b21(t) x − b22(t) y) ,
introducing suitable average conditions. Marva et al.[9] use invariant regions, varying in t, to prove the existence of periodic solutions of the predator-prey system x0 = λ(t) x(1 − x k(t)) − a(t) xy b(t) + x y0= −y(µ(t) + e(t)y) +α a(t) xy
Zhu and Wang[14] consider the modified Leslie-Gower system x0 = x(r1(t) − b(t) x − a1(t) y x + k1 ) y0= y(r2(t) − a2(t) y x + k2 ) . (1.6)
Their arguments for the stability of the periodic solution take advantage from the presence of k2 > 0 in the second equation of (1.6) and do not work for
system (1.3).
In section 3, we establish sufficient average criteria for the permanence of so-lutions and the existence of positive periodic soso-lutions of (1.3) determining an invariant region, depending on t, as in [9]. In section 4 we present sufficient con-ditions guaranteeing the global stability of the unique positive periodic solution (u∗(t), v∗(t)) of (1.3). As the first step of our technique, we transform model (1.3) into the differential system (4.2) in which the periodic solution becomes the origin. Then we introduce a Lyapunov function, different from (1.5), which better suits the specific model (1.3). Indeed, its derivative along the solutions of (4.2) leads to a sort of quadratic form whose properties are crucial for the application of the Lyapunov method. Inspired by [8], in Theorem 4.1, the suffi-cient conditions for the global attractivity of (u∗(t), v∗(t)) are given via integral averages. Even the stronger pointwise assumptions (4.11) and (4.12) are less restrictive than those required in [10] (see the conclusive discussion in Section 5), thus Theorem 4.1 and Corollary 4.1 improve previous results concerning pe-riodic Holling-Tanner systems. Moreover, Corollary 4.1, when applied to the autonomous system (2.1), gives the known global stability condition m > (bγ), as Corollary 4.2 shows.
2
The autonomous system
For convenience of the reader, in this section, we state some results concerning the Holling-Tanner model
u0= u(a − u − v u + m) v0= v(b − v γ u) , (2.1)
with a, b, γ, m > 0 and initial condition u(0) > 0, v(0) > 0. We are only interested in the dynamics of positive solutions to (2.1). System (2.1) has a unique positive equilibrium (u∗, v∗), where
u∗=a − m − b γ +p(a − m − b γ)
2+ 4a m
2 , v
∗= (b γ)u∗.
(u∗, v∗) is the intersection point, in the plane u v, between the isocline curves v = (a − u)(u + m) and v = (b γ)u. Hence u∗ is the positive solution of the equation
It is well-known, that (u∗, v∗) is asymptotically stable if u∗≥a − m
2 .
A simple sufficient condition for the global stability of the equilibrium in the first quadrant can be easily derived from [6], as the following theorem shows. Theorem 2.1 If
(bγ) < m , (2.2) then (u∗, v∗) is globally asymptotically stable.
Proof. If (2.2) holds, we get
0 = a − u∗− v ∗ u∗+ m > a − u ∗−v∗ u∗ = a − u ∗− (b γ) > a − u∗− m . Then u∗> a − m ,
and (u∗, v∗) is asymptotically stable. In [6], by the Lyapunov method, it is proved that (u∗, v∗) is globally stable for system (2.1) if the horizontal line v = v∗ divides the parabola v = (a − u)(u + m) into two disjoint parts for 0 < u < a. Since the parabola intersects the v-axes at v = a m, previous condition is verified if inequality
m a > v∗ holds. Under hypothesis (2.2) we deduce
m > (b γ)(u
∗
a ) = v∗
a , and the required global stability is proved. 2
3
Periodic solutions
For a continuous, T-periodic function f (t), we denote its integral average (or mean value) by [f (t)] = 1 T Z T 0 f (t) dt .
In order to achieve the existence of periodic solutions, we first note that, in absence of predators (v(t) = 0), system (1.3) is reduced to the logistic equation
u0= u(a(t) − u) .
Lemma 3.1 [1] Let p(t), q(t) be continuous and T-periodic functions, q(t) > 0 and [p(t)] > 0. Then the logistic equation
x0= x(p(t) − q(t)x)
admits a unique positive, T-periodic solution x∗(t). Moreover, if x(t) is any other positive solution, one yields
lim
t→+∞|x(t) − x
∗(t)| = 0 .
If (u(t), v(t)) is a solution to (1.3) with initial condition u(0), v(0) > 0, then u(t), v(t) > 0 for all t > 0 .
Such solutions will be called positive.
The next theorem gives a permanence result for predator-prey model (1.3). Henceforth we assume a(t), b(t), m(t), γ(t) are continuous T-periodic functions with [a(t)], [b(t)] > 0 and m(t), γ(t) > 0.
Denote byeu(t) the positive T-periodic solution of the logistic equation
x0= x(a(t) − x) (3.1) and byv(t) the positive T-periodic solution of the logistic equatione
y0= y(b(t) − y
γ(t)u(t)e ) . (3.2) Theorem 3.1 If the inequality
[a(t)] > e v(t) m(t) (3.3)
holds, then, for any positive solution (u(t), v(t)) of (1.3), there exists ¯t > 0 such that
u(t) ≤ u(t) ≤u(t),e v(t) ≤ v(t) ≤ev(t), for t > ¯t, where u(t) is the positive periodic solution of the logistic equation
x0= x((a(t) − ev(t) m(t)) − x) and v(t) is the periodic solution to
y0= y(b(t) − y γ(t) u(t)) .
Proof. Our mail tool is the comparison theorem for differential equations. Let (u(t), v(t)) be a positive solution of (1.3). Since
u0(t) ≤ u(t)(a(t) − u(t)), [a(t)] > 0 ,
by Lemma 3.1, for all positive solutions of equation (3.1), we have lim
From the comparison theorem one deduces the existence of t0> 0 such that
u(t) ≤eu(t), t > t0.
As a consequence, for t > t0,
v0(t) ≤ v(t)(b(t) − v(t) γ(t)u(t)e ) .
Using the assumption [b(t)] > 0 for equation (3.2) and arguing as before, we can say that
v(t) ≤ev(t), t > t1,
where t1> t0 is an appropriate value of time. For t > t1, we get
u0(t) ≥ u(t)((a(t) − ev(t)
m(t)) − u(t)) .
From the above arguments and (3.3), it follows the existence of t2 > t1 such
that
u(t) ≥ u(t), t > t2,
so that
v0(t) ≥ v(t)(b(t) − v(t)
γ(t) u(t)), t > t2. Thus, for a suitable ¯t > t2, the inequality
v(t) ≥ v(t), t > ¯t is verified. The proof is complete. 2
Theorem 3.2 Suppose that inequality (3.3) is satisfied, then system (1.3) has at least a positive, T-periodic solution (u∗(t), v∗(t)) such that
u(t) ≤ u∗(t) ≤eu(t), v(t) ≤ v∗(t) ≤ev(t), t ∈ [0, T ] .
Proof. The periodic functions u(t), v(t),u(t),e ev(t) are defined in the previous theorem.
Set
K = [u(0),u(0)] × [v(0),e ev(0)] ,
and let (u(t), v(t)) be a solution to (1.3) with initial condition (u(0), v(0)) ∈ K. Since u(0) ≤u(0) ande
u0(t) ≤ u(t)(a(t) − u(t)) ,
we deduce that u(t) ≤eu(t) for t > 0. Repeating this arguments as in Theorem 3.1, we can state
u(0) = u(T ) ≤ u(T ) ≤eu(T ) =eu(0) , v(0) = v(T ) ≤ v(T ) ≤ev(T ) =v(0) ,e
thus
(u(T ), v(T )) ∈ K . Consider the map
F : K −→ K, F (x, y) = (u(T ), v(T )) ,
where (u(t), v(t)) is the solution of (1.3) with initial condition u(0) = x, v(0) = y. By the previous argument, we get
F (x, y) ∈ K ,
so F is well-defined. Since F is continuous, the Brouwer fixed point theorem ensures the existence of a point (bu,bv) ∈ K such that
F (u,b bv) = (bu,bv) .
By construction, the solution (u∗(t), v∗(t)) with initial condition (u,b bv) verifies ((u∗(0), v∗(0)) = ((u∗(T ), v∗(T )) .
The properties of periodic differential equations[3] guarantee that (u∗(t), v∗(t))
is the searched periodic solution to (1.3). 2
Previous Theorems 3.1 and 3.2, when applied to autonomous system (2.1), give the existence of an invariant region.
Corollary 3.1 Suppose that, for system (2.1), inequality (2.2) holds and put u = (1 −b γm)a. Then the rectangle
R = [u, a] × [(b γ)u, (b γ)a] is invariant and attractive.
Proof. It is immediate that
e
u(t) = a, ev(t) = (b γ)a .
In this autonomous case, u(t) = u is the positive equilibrium of the logistic equation
x0= x((a −b γ a
m ) − x) with (1 − b γ
m)a > 0 . Consequently, v(t) = (b γ)u is the positive equilibrium of
y0 = y(b − v γ u) .
4
Global stability
For the sake of convenience we give the following definition.
Definition 4.1 Let (u∗(t), v∗(t)) be a positive T-periodic solution of system (1.3). We say that it is globally asymptotically stable (or globally attractive) if any other positive solution (u(t), v(t)) of (1.3) has the property
lim
t→+∞|u(t) − u
∗(t)| = 0 = lim
t→+∞|v(t) − v ∗(t)| .
Lemma 4.1 Let (u∗(t), v∗(t)) be a positive periodic solution of system (1.3). Under the substitution
x(t) = u(t)
u∗(t)− 1, y(t) =
v(t)
v∗(t)− 1 , (4.1)
system (1.3) turns into x0= (1 + x) −u∗(t) x − v∗(t) y u∗(t) (x + θ(t))+ v∗(t) u∗(t) θ(t)· x (x + θ(t)) y0= (1 + y) v ∗(t) γ(t) u∗(t) − y x + 1 + x x + 1 , (4.2)
where θ(t) denotes the following periodic function
θ(t) = m(t)
u∗(t)+ 1 . (4.3)
Proof. Using (4.1), the first equation of differential system (1.3) becomes
x0 = u 0 u∗(t)− u (u∗(t))0 (u∗(t))2 = u u∗(t) −u + u∗(t) − v u + m(t)+ v∗(t) u∗(t) + m(t) = (1 + x) −u∗(t) x −v ∗(t) u∗(t) y + 1 x + θ(t) − 1 θ(t) . Since y + 1 x + θ(t)− 1 θ(t) = θ(t) y − x θ(t)(x + θ(t)) , we obtain x0 = (1 + x) −u∗(t) x − v ∗(t) y u∗(t) (x + θ(t)) + v∗(t) u∗(t) θ(t)· x x + θ(t) .
Analogous calculations yield
y0= v 0 v∗(t)− v (v∗(t))0 (v∗(t))2 = v v∗(t) − v γ(t) u+ v∗(t) γ(t) u∗(t) = −(y + 1) v ∗(t) γ(t) u∗(t) y + 1 x + 1 − 1 = (1 + y) v ∗(t) γ(t) u∗(t) − y x + 1 + x x + 1
and (4.2) is proved. 2
From Theorem 3.1, for any positive solution (u(t), v(t)), for t > ¯t, we get x(t) = u(t) u∗(t)−1 ≤ x(t) ≤ e u(t) u∗(t)−1 =ex(t), y(t) = v(t) v∗(t)−1 ≤ y(t) ≤ e v(t) v∗(t)−1 =y(t) .e (4.4) Then (x(t), y(t)) ∈ Q(t), t > ¯t , where, for each t, Q(t) is the following rectangle
Q(t) = [x(t),ex(t)] × [y(t),y(t)]e
that varies with time. Under condition (3.3), fix a positive periodic solution (u∗(t), v∗(t)) of (1.3) and put
α = m(t) u∗(t)
, θα= α + 1 . (4.5)
Note that θα= [θ(t)]. Let G(t; x, y) be the function defined by
G(t; x, y) = − u∗(t)(x + θα) − v∗(t) u∗(t) · (x + θα) θ(t)(x + θ(t)) x2 (4.6) +v ∗(t) u∗(t) σ γ(t)− x + θα x + θ(t) x y − σ v ∗(t) γ(t) u∗(t)y 2,
where θ(t) and θα are given by (4.3) and (4.5), respectively. In the next
theo-rem a suitable Lyapunov function will be introduced by means of the following function H(x, y) = Z x+1 1 (1 −1 s)(1 + α s) ds + σ Z y+1 1 (1 −1 t) dt, σ = [γ(t)] . (4.7) It easy to see that the function H(x, y) is continuous in ] − 1, +∞[×] − 1, +∞[, H(0, 0) = 0 and H(x, y) is positive for all other value of (x, y).
Theorem 4.1 Assume that condition (3.3) holds and let (u∗(t), v∗(t)) be a pos-itive periodic solution of (1.3). Such solution is globally attractive if
[Λ(t)] < 0 , (4.8) where, for each t > 0,
Λ(t) = max
(x,y)∈Q(t)
G(t; x, y)
(x + 1)H(x, y) . (4.9) Proof. The proof is based on a suitable Lyapunov function. Let (u(t), v(t)) be a solution of (1.3) and (x(t), y(t)) the corresponding solution of system (4.2) under substitution (4.1). Introduce the following Lyapunov function
where H(x, y) is given by (4.7). Since, for each t, the solutions of (4.2) ultimately enter the set Q(t), we restrict the study for this set. Using Lemma 4.1, (4.5) and (4.7), the time derivative of V (t) leads to
V0(t) = x x + 1 · x + θα x + 1 x 0(t) + σ y y + 1y 0(t) = 1 x + 1[x(x + θα)(−u ∗(t)x −v∗(t) u∗(t)· y x + θ(t)+ v∗(t) u∗(t)θ(t) · x x + θ(t)) + σ γ(t)( v∗(t) u∗(t)x y − v∗(t) u∗(t)y 2)] ,
so that, using (4.6), the previous equality can be written in the form
V0(t) = G(t; x(t), y(t)) x(t) + 1 .
For a fixed t > ¯t, let us investigate the behavior of the ratio G(t; x, y) (x + 1)H(x, y) , in a neighbourhood of (0, 0). Since H(x, y) = (x − log(x + 1)) + α(log(x + 1) − x x + 1) + σ(y − log(y + 1)) , we have H(x, y) =x 2 2 + α x2 2 + σ y2 2 + o(x
2+ y2). Moreover, one can easily verify
that
G(t; 0, y) < 0 for y 6= 0, G(t; 0, 0) = 0 and, for each h ∈ R,
lim
x→0
G(t; x, h x) (h x + 1)H(x, h x) is finite. The previous argument shows that G(t; x, y)
(x + 1)H(x, y) is bounded near the origin. Consequently, it makes sense to consider the function Λ(t) defined by (4.9). For each fixed t > ¯t
V0(t) V (t) =
G(t; x(t), y(t))
(x(t) + 1)H(x(t), y(t)) ≤(x,y)∈Q(t)max
G(t; x, y) (x + 1)H(x, y) , that is
V0(t) ≤ Λ(t) V (t), t > ¯t . Using assumption (4.8), we get
lim
t→+∞V (t) = 0 ,
which implies
lim
Going back to (u(t), v(t)) through (4.1), we deduce lim t→+∞|u(t) − u ∗(t)| = 0 = lim t→+∞|v(t) − v ∗(t)| ,
and the global attractivity of (u∗(t), v∗(t)) is proved. 2
The next corollary provides pointwise conditions ensuring the same conclusion of the previous theorem. Let us introduce a new notation. For any fixed t > 0, put ∆(t; x) = v ∗(t) u∗(t) 2 σ γ(t)− x + θα x + θ(t) 2 (4.10) −4 σ v ∗(t) γ(t) u∗(t) u∗(t)(x + θα) − v∗(t) u∗(t)· (x + θα) θ(t)(x + θ(t)) , x > −1 . Corollary 4.1 Under (3.3), let (u∗(t), v∗(t)) be a positive periodic solution of (1.3). Suppose
v∗(t)
u∗(t) ≤ m(t), t ∈ [0, T ] (4.11)
and
∆(t; x) < 0 for all t > 0, x ∈ [x(t),ex(t)] , (4.12) where x(t) andx(t) are defined in (4.4). Then (ue ∗(t), v∗(t)) is globally asymp-totically stable.
Proof. Fix t > 0. Using the notations of the previous theorem, our starting step is the equality
V0(t) = G(t; x(t), y(t)) x(t) + 1 .
Thus V0(t) < 0 if G(t; x, y) is negative for all (x, y) ∈ Q(t). Now look at G(t; x, y) as a quadratic form in variables x, y. Since x + θα= x + 1 + α > 0, in
(4.6) the coefficient of x2 is negative iff
u∗(t) −v
∗(t)
u∗(t)·
1
θ(t)(x + θ(t)) > 0 . (4.13) Taking into account (4.3) and (4.4), one gets
x + θ(t) ≥ u(t) + m(t) u∗(t) and u∗(t) 1 − v ∗(t) (u∗(t))2 · 1 θ(t)(x + θ(t)) ≥ u∗(t) 1 − v ∗(t) u∗(t)· u∗(t) (u∗(t) + m(t))(u(t) + m(t)) > u∗(t) 1 − v ∗(t) u∗(t)· 1 u(t) + m(t) .
Using assumption (4.11), 1 − v ∗(t) u∗(t)· 1 u(t) + m(t) ≥ 1 − m(t) u(t) + m(t) > 0 ,
so that (4.13) holds. Using (4.10), (4.13) and hypothesis (4.12), one yields G(t; x, y) < 0 for any (x, y) ∈ Q(t) .
Therefore
Λ(t) < 0 , and the conclusion of Theorem 4.1 follows. 2
Our approach yields Theorem 2.1 when applied to system (2.1).
Corollary 4.2 For autonomous system (2.1), condition (2.2) is sufficient for the global asymptotic stability of equilibrium (u∗, v∗).
It easy to see that (3.3) becomes
a > (bγ)a m
which is true by (2.2). Moreover from (2.2) we deduce v∗
u∗ = (b γ) < m ,
so that (4.11) is satisfied. It remains to check the validity of (4.12). Since all coefficients of (1.3) are constant, we have that θα = θ(t) = θ = 1 +
m u∗ and σ = γ(t) = γ. Consequently ∆(t; x) = ∆(x) = −4v ∗ u∗(u ∗(x + θ) − v∗ u∗θ) .
Observe that, by Corollary 3.1, x > x(t) = u
u∗ − 1. Then u∗(x + θ) − v ∗ u∗θ ≥ u + m − (bγ)u∗ u∗+ m > u + m − m u∗ u∗+ m > 0 , which implies ∆(x) < 0 . The application of Corollary 4.1 ends the proof. 2
The result of the previous corollary agrees with the conclusions in [5] for the corresponding diffusive system.
5
Conclusion and example
In this paper we have investigated the global behavior of a known predator-prey system, called Holling-Tanner model, assuming continuous and periodic coeffi-cients. As first result, we have shown the existence of an invariant and attractive region in which the positive solution is located. Our main tools are the prop-erties of the logistic equation, the comparison theorem and the Brouwer fixed point theorem. Afterwords, we have provided sufficient conditions ensuring that all positive solutions of our system tend to the periodic solution (u∗(t), v∗(t)), as the time goes to infinity. By a suitable change of variables, our model has been transformed into a differential system for which the periodic solution becomes (0, 0). This circumstance is convenient for eventual numerical simulations. The method of Lyapunov has been employed to obtain the global asymptotic stabil-ity of (u∗(t), v∗(t)).
As an application to the theoretical results, we consider the following 2π−periodic system u0 = u (a(t) − u − v u + m(t)) v0= v (2 − 7v 8u ) , (5.1)
with initial conditions u(0) > 0, v(0) > and
a(t) = 5 − 0.2 cos(t), m(t) = 3 + 0.8 cos(t) 1 − 0.2 cos(t) . Note that min t∈[0,2π] m(t) = 11 6 .
Letu(t) be the positive periodic solution to the logistic equatione u0 = u (a(t) − u) ,
hence [u(t)] = [a(t)] = 5 and maxe eu(t) ≤ max a(t) = 5.2.
According to the notations of Section 3,ev(t) denotes the positive periodic solu-tion to the logistic equasolu-tion
v0 = v (2 − 7v 8u ) , hence e v(t) e u(t) = 7
4. Condition (3.3) is satisfied because e v(t) m(t) = e v(t) e u(t)· e u(t) m(t) ≤ e v(t) e u(t) · maxu(t)e min m(t)= 7 4 · 5.2 11/6 < 5 = [a(t)] . It easy to verify that system (5.1) has
as periodic solution. To investigate its global stability we apply Corollary 4.1. Since v∗ u∗ = 7 4 < 11 6 = min m(t) ≤ m(t), t ∈ [0, 2π] ,
condition (4.11) is satisfied. It remains to verify (4.12). A direct calculation leads to ∆(t; u) = 7 4 " 7 4 1 −u + [m(t)] u + m(t) 2 − 4 u + [m(t)] − 7u + [m(t)] 4 + m(t) #
where u = u∗(x + 1) = 4(x + 1). Since ∆(t; u) attains its maximum value at u = 0 (for each fixed t and u ≥ 0), it is enough to verify that
7 4 1 − [m(t)] m(t) 2 − 4[m(t)] 1 − 7 m(t)(4 + m(t)) < 0 .
Previously inequality can be checked by sample numerical calculations. There-fore (4.12) holds, too. We conclude that (u∗, v∗) is globally attractive.
Among the cited references, the only paper presenting results which may be directly compared to ours, is [10]. It is possible to check that the required assumption for the global asymptotic stability (see Theorem 3.5[10]) are not satisfied by system (5.1). For instance, one of such conditions is the estimate
(min a(t)) · (min m(t)) > b γ (max a(t)) that is
4.8 ·11 6 >
7 4· 5.2 ,
but previous inequality is obviously false. Then our example cannot be treated by the investigation developed in [10]. Finally, in view of the above discussion, we remark that our improvement of the results in [10] are due to the new approach used to attack the problem.
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