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SOME RESULTS ON PREDATOR-PREY DYNAMIC SYSTEMS WITH BEDDINGTON-DEANGELIS TYPE FUNCTIONAL RESPONSE ON TIME SCALE CALCULUS.

1. Introduction

The subject of mathematical ecology in biomathematics is the relationship between species and outer environment. Moreover, the connections between different species describe predator-prey dynamic systems. In this type of dynamic sytems, most important issues are global attractivity and permanence. Global attractivity shows the stability of the species in their circumstances and permanence shows whether the considered species are permanent against environmental conditions. The conditions that make different predator-prey dynamic systems permanent and globally attractive were studied in [4], [9] and [18]. Another important notion in predator-prey systems is functional response, which explains the effect of predator on prey and vice versa. Therefore, various types of functional responses such as semi-ratio dependent, Holling-type functional responses have been investigated in several studies like [8], [11], [16], [17], [19].

In this paper, we consider the predator-prey systems with Beddington DeAngelis type functional response. This type of functional response was first considered in [1] and [7] by Beddington and DeAngelis respectively. At low densities, this type of functional response can avoid singular behavior of ratio-dependent models. Also predator feeding can be described by this functional response much better over a range of predator-prey abundances. Because of these advantages of the Beddington-DeAngelis type functional response, we have preferred to study on that system. For such kind of systems, boundedness of solution, permanence and global attractivity are important topics for the mathematical analysis which give information about the future of the population of the species.

On the other hand, when the size of the population is rarely small or has non- overlaping generation, then discrete models are more appropriate than continuous ones. Since time scale models unify discrete and continuous models, this type of models becomes more applicable to real life than the others. Many studies about global attractivity and permanence of the predator-prey dynamic systems with Beddington-DeAngelis functional response on continuous and discrete cases have been done. [6] and [14] are some of the examples for continuous case and [9] is the example for discrete case. Additionally, some examples on predator-prey dynamic systems for time scale case are [3], [10] and [15].

What we study in this paper is permanence and global attractivity of the solutions for general time scales case of the predator-prey system with Beddington- DeAngelis type functional response. We have found some conditions for permanence and global attractivity of the considered system. It enables us to make analysis about future of the species.

2. Preliminaries

By a time scale, denoted by T, we mean a nonempty closed subset of R. The theory of time scales give a way to unify continuous and discrete analysis. The followings are some important notions about the time scales calculus which are taken from [2] and [13].

The set [T.sup.[kappa]] is defined by [T.sup.[kappa]] := T/([rho](sup T), sup T] and the set [T.sub.[kappa]] is defined by [T.sub.[kappa]] := T/[inf T, [sigma][paragraph](inf T)). The backward jump operator [rho] : T [right arrow] T is defined by [rho](t) := sup[(-[infinity], t).sub.T], for t [member of] T. The forward jump operator [sigma] : T [right arrow] T is defined by [sigma](t) := inf[(t, [infinity]).sub.T], for t [member of] T. Here, it is assumed that inf 0 = sup T and sup 0 = inf T.

For a function f : T [right arrow] T, we denote the [DELTA]-derivative of f at t [member of] [T.sup.[kappa]] as [f.sup.[DELTA]](t) and it is defined as follows: For all [epsilon] > 0, there exists a neighborhood U [subset] T of t [member of] [T.sup.[kappa]] such that

[absolute value of (f([sigma](t)) - f(s) - [f.sup.[DELTA]](t)([sigma](t) - s))] [less than or equal to] [epsilon]|[sigma](t) - s)]

for all s [member of] U.

A function f : T [right arrow] R is rd-continuous if it is continuous at right dense points in T and its left-sided limits exist at left-dense points in T. The class of real rd- continuous functions defined on a time scale T is denoted by [C.sub.rd](T, R). If f [member of] [C.sub.rd](T, R), then there exists a function F(t) such that [F.sup.[DELTA]](t) = f (t). The delta integral is defined by [[integral].sup.b.sub.a] f (x)[increment of x] = F(b) - F(a).

3. Predator-Prey Dynamic System with Beddington DeAngelis Type Functional Response

We investigate the following equation:

(3.1) [mathematical expression not reproducible]

In this sytem;

1. a(t) - b(t) exp(x(t)) is the specific growth rate of the prey in the absense of predator.

2. d(t) is the death rate of predator.

3. -c(t) exp(y(t))/[alpha](t) + [beta](t) exp(x(t)) + m(t) exp(y(t)), is the Beddington DeAngelis type effect of predator on prey.

4. f(t) exp(x(t))/[alpha](t) + [beta](t) exp(x(t)) + m(t) exp(y(t)) is the Beddington DeAngelis type effect of prey on predator.

Consider the following system:

(3.2) [mathematical expression not reproducible].

The following information is taken from [14]. Let T = R, then in (3.1), by taking exp(x(t)) = x(t) and exp(y(t)) = y(t), we obtain the equality (3.2), which is the standart predator-prey system with Beddington-DeAngelis functional response governed by ordinary differential equations. Many studies have been done on this system (see [5], [6] and [12]).

Let T = Z, by using equality (3.1), we get

[mathematical expression not reproducible].

Here, again by taking [mathematical expression not reproducible], we obtain

(3.3) [mathematical expression not reproducible],

which is the discrete time predator-prey system with Beddington-DeAngelis type functional response and also the discrete analogue of (3.2). This system was studied in [9], [21] and [20]. Since (3.1) incorporates (3.2) and (3.3) as special cases, we call (3.1) the predator-prey dynamic system with Beddington DeAngelis functional response on time scales.

For equation (3.1), exp(x(t)) and exp(y(t)) denote the density of prey and the predator. Therefore x(t) and y(t) could be negative. By taking the exponentials of x(t) and y(t), we obtain the number of preys and predators that are living per unit of an area. In other words, for the general time scale case, our equation is based on the natural logarithm of the density of the predator and prey. Hence, x(t) and y(t) could be negative.

For equations (3.2) and (3.3), since [mathematical expression not reproducible], the given dynamic systems directly depend on the density of the prey and predator.

4. Permenance

Taking [mathematical expression not reproducible] in (3.1), then we get

(4.1) [mathematical expression not reproducible]

Assume a(t), b(t), c(t), d(t), f(t), [beta](t), m(t) > 0 and [alpha](t) [greater than or equal to] 0. Also suppose that these functions are bounded from above. Each of them is from [C.sub.rd](T, R) and x(t),y(t) [member of] [C.sub.rd](T, R). Additionally, [sup.sub.t[member of]T] a(t) = [a.sup.u], [inf.sub.t[member of]T] a(t) = [a.sup.l]. Similar representations are used for supremum and infimum of the other coefficient functions of system (4.1).

Definition 4.1. System (4.1) is called permanent if there exists positive constants [r.sub.1], [r.sub.2], [R.sub.1], and [R.sub.2] such that solution (x(t), y(t)) of system (4.1) satisfies

[mathematical expression not reproducible].

Lemma 4.2. If - [[integral].sup.w.sub.0] d(t)[DELTA]t + [[integral].sup.w.sub.0] f(t)/[beta](t) [DELTA]t < 0, then for all positive solutions of system (4.1), exp(y(t)) tends to 0 as t tends to infinity in system (3.1).

Proof. Integrate

[y.sup.[DELTA]](t) = -d(t) + f(t) exp(x(t))/[alpha](t) + [beta](t) exp(x(t)) + m(t) exp(y(t)) [less than or equal to] - d(t) + d(t)/[beta](t)

from 0 to t. Then, we get

y(t) [less than or equal to] y(0) + [[integral].sup.t.sub.0] -d(s) + f(s)/[beta](s) [DELTA]s.

Thus,

exp(y(t)) [less than or equal to] exp(y(0)) exp ([[integral].sup.t.sub.0] -d(s) + f(s)/[beta](s) [DELTA]s).

From the assumption, we have [lim.sub.t[right arrow][infinity]] exp ([[integral].sup.t.sub.0] -d(s) f(s)/[beta](s) [DELTA]s) = 0. Hence, [lim.sub.t[right arrow][infinity]] exp(y(t)) = 0.

Remark 4.3. If (4.1) satisfies conditions of Lemma 4.2, then the system can not be permanent by Definition 4.1.

Lemma 4.4. If conditions for the coefficient functions of system (4.1) are satisfied, then [??](t) [less than or equal to] [a.sup.u]/[b.sup.l] exp([mu][a.sup.u]) := [G.sub.1]. In addition to conditions on the coefficient functions of system (4.1) if -[d.sup.l] + [f.sup.u]/[[beta].sup.l] [greater than or equal to] 0 is satisfied, we have the following

[??](t)t [less than or equal to] [f.sup.u][G.sub.1]/[d.sup.l][m.sup.l] exp ([mu](- [d.sup.l] + [f.sup.u]/[[beta].sup.l])) := [G.sub.2],

where [mu] = [max.sub.t[member of]T] [mu](t)

Proof. Let us start with the first equation of (4.1),

(4.2) [mathematical expression not reproducible].

Set [M.sub.1] := [a.sup.u]/[b.sup.l](k + 1), where 0 < k < exp{[mu][a.sup.u]} - 1. If [??](t) is not oscillatory about [M.sub.1], there exists [T.sub.1] > 0 such that [??](t) > [M.sub.1] for t > [T.sub.1] or [??](t) < [M.sub.1] for t > [T.sub.1].

If [??](t) < [M.sub.1] for t > [T.sub.1], then [??(t) [less than or equal to] [a.sup.u]/[b.sup.l] exp{[mu][a.sup.u]}. If [??](t) > [M.sub.1] for t > [T.sub.1], then [(ln([??](t))).sup.[DELTA]] [less than or equal to] -k[a.sup.u]. Hence, there exists [T.sub.2] = [T.sub.1] + [tau] such that for t > [T.sub.2], [??](t) < [M.sub.1], which is a contradiction.

If [??](t) is oscillatory about [M.sub.1] for t > [T.sub.1] and aft) be an arbitrary local maximum of ln([??](t)), then

[mathematical expression not reproducible].

Therefore [mathematical expression not reproducible]. If t is right dense, then [mathematical expression not reproducible]. If t is right scattered, by integrating first equation of (4.1) from t to a(t) and using (4.2), we obtain

[mathematical expression not reproducible]

and

(4.3) [mathematical expression not reproducible].

Since [sigma]([??]) be an arbitrary local maximum of ln([??](t)), then lim [sup.sub.t[right arrow][infinity]] [??](t) [less than or equal to] [G.sub.1].

Hence, lim [sup.sub.t[right arrow][infinity]] x(t) [less than or equal to] [R.sub.1].

Consider the second equation of (4.1), we get

(4 4) [mathematical expression not reproducible].

Define [M.sub.2] := [f.sup.u][G.sub.1]/[d.sub.l][m.sub.l] (k + 1), where 0 < k < exp ([mu](-dl + [f.sup.u]/[[beta].sup.l])) - 1. If [??](t) is not oscillatory about [M.sub.2], there exists [T.sub.3] > 0 such that [??](t) > [M.sub.2] for t > [T.sub.3] or [??] (t) < [M.sub.2] for t > [T.sub.3]. If [??] (t) < [M.sub.2] for t > [T.sub.3], then [??] (t) < [f.sup.u][G.sub.1]/[d.sup.l][m.sup.l] (k + 1). If [??] (f) > [M.sub.2] for i > [T.sub.3], then [(ln([??](t))).sup.[DELTA]] [less than or equal to] -k [f.sup.u][G.sub.1]/[[alpha].sup.u] + [[beta].sup.u][G.sub.1] + [m.sup.u] [M.sub.2]. Hence, there exists [T.sub.4] = [T.sub.3] + [tau] such that for t > [T.sub.4], [??](t) < [M.sub.2], which is a contradiction.

If [??](t) is oscillatory about [M.sub.2] for t > [T.sub.3], let [sigma]([??]) be an arbitrary local maximum of ln([??](t)), then by using second equation of (4.1), we can conclude that

[mathematical expression not reproducible].

Therefore, [mathematical expression not reproducible]. If t is right dense, then [mathematical expression not reproducible].

If [??] is right scattered, integrate (4.4) from [??] to [sigma]([??]) for the same w above and we obtain

[mathematical expression not reproducible].

(4.5) [mathematical expression not reproducible].

Since [sigma]([??]) be an arbitrary local maximum of ln([??](t)), then lim [sup.sub.t[theta][infinity]] [??](t) [less than or equal to] [G.sub.2]. Hence lim [sup.sub.t[right arrow][infinity]] y(t) [less than or equal to] [R.sub.2].

Remark 4.5. For all solutions of system (3.1), if exp(y(t)) does not tend to 0 as t tends to infinity, Lemma 4.4 follows from Lemma 4.2.

Lemma 4.6. For (4.1) when [??](t) [less than or equal to] [G.sub.1], [a.sup.l] - [b.sup.u][G.sub.1] - [c.sup.u]/[m.sup.l] [less than or equal to] 0 and [a.sup.l] - [c.sup.u][m.sub.l] [greater than or equal to] 0 are satisfied, then

[mathematical expression not reproducible]

and when [mathematical expression not reproducible] are satisfied, then

[mathematical expression not reproducible],

where [mu] = [max.sub.t[member of]T] [mu](t).

Proof. Consider the first equation of (4.1) and we obtain

(4.6) [mathematical expression not reproducible].

If [a.sup.l] - [b.sup.u][G.sub.1] - [c.sup.u]/[m.sup.l] > 0, then there exists [??] such that [mathematical expression not reproducible], for t > [??]. So, there is a contradiction. Therefore [a.sup.l] - [b.sup.u][G.sub.1] - [c.sup.u]/[m.sup.l] [less than or equal to] 0.

Take [N.sub.1] = 1/[b.sup.u]([a.sup.l] - [c.sup.u]/[m.sup.l])(1 - [??]), where [??] = 1 - exp (w ([a.sup.l] - [c.sup.u]/[m.sup.l] - [b.sup.u][G.sub.1])). Suppose that [??](t) is not oscillatory around [N.sub.1]. Then there exists [T.sub.5], such that [??](t) > [N.sub.1] for t > [T.sub.5] or [??](t) < [N.sub.1] for t > [T.sub.5]. If [??](t) > [N.sub.1] for t > [T.sub.5], then [??](t) satisfies the desired result. Since [??] [greater than or equal to] 0, then the condition [a.sup.l] - [c.sup.u]/[m.sup.l] [greater than or equal to] 0 must be satisfied. If [??](t) < [N.sub.1] for t > [T.sub.5], then [mathematical expression not reproducible]. Since ([a.sup.l] - [c.sup.u]/[m.sup.l])[??] > 0, there exists [T.sub.6] such that for t > [T.sub.6], we have [??](t) > [N.sub.1] which is a contradiction. Suppose that [??](t) is oscillatory around [N.sub.1] and [sigma]([t.sub.1]) be an arbitrary local minimum of ln([??](t)), thus

[mathematical expression not reproducible].

Thus, we have,

1/[b.sup.u] ([a.sup.l] - [c.sup.u]/[m.sup.l]) [less than or equal to] 1/b([t.sub.1]) (a([t.sub.1]) - c([t.sub.1])/m([t.sub.1])) [less than or equal to] [??]([t.sub.1]).

If [t.sub.1] is right dense, then [??]([sigma]([t.sub.1])) [greater than or equal to] 1/[b.sup.u]([a.sup.l] - [c.sup.u]/[m.sup.l]). If [t.sub.1] is right scattered, the following is obtained by integrating (4.6) from [t.sub.1] to [sigma]([t.sub.1])

[mathematical expression not reproducible].

Since [??]([sigma]([t.sub.1])) is the arbitrary local minimum, then [mathematical expression not reproducible], i.e. lim [sup.sub.t[right arrow][infinity]] x(t) [less than or equal to] [r.sub.1].

Considering the second equation of (4.1), we have

(4.7) [mathematical expression not reproducible].

If [mathematical expression not reproducible], then there exits [T.sub.7] such that for t > [T.sub.7], one can see [??](t) > [G.sub.2] which is a contradiction. Thus [mathematical expression not reproducible].

Let us take [N.sub.2] such that [N.sub.2] = 1/[d.sup.u][m.sup.u] (- [D.sup.u][[alpha].sup.u] - [d.sup.u][[beta].sup.u] [[??].sub.1] + [f.sup.l] [[??].sub.1])(1 - r) where [mathematical expression not reproducible]. Assume [??](t) is not oscillatory around [N.sub.2]. Then, there exists [T.sub.8], such that for t > [T.sub.8] [??](t) > [N.sub.2] or [??](t) < [N.sub.2].

For the first case,

[mathematical expression not reproducible].

Since [??] > 0, then -[d.sup.u][a.sup.u] - [d.sup.u][[beta].sup.u][[??].sub.1] + [f.sup.l][[??].sub.1] [greater than or equal to] 0.

For the second case

[mathematical expression not reproducible].

Since -[d.sup.u][a.sup.u] - [d.sup.u][[beta].sup.u][[??].sub.1] + [f.sup.l][[??].sub.1] > 0, there exists [T.sub.9], such that [??] (t) > [N.sub.2] for t > [T.sub.9], which is a contradiction.

Assume [??](t) is oscillatory around [N.sub.2] and [sigma]([t.sub.2]) be an arbitrary local minumum of ln([??](t)), then by (4.7) we have

[mathematical expression not reproducible].

So, we get [mathematical expression not reproducible]. Thus, [mathematical expression not reproducible].

If [t.sub.2] is right dense, we have [mathematical expression not reproducible]. If [t.sub.2] is right scattered, by integrating (4.7) from [t.sub.2] to [sigma]([t.sub.2]) we obtain

[mathematical expression not reproducible].

By the above inequality, we have

[mathematical expression not reproducible].

Since [??]([sigma]([t.sub.2])) is the arbitrary local minimum, then [mathematical expression not reproducible], i.e. lim [sup.sub.t[right arrow][infinity]] y(t) < [r.sub.2].

If (3.1) satisfies all the conditions of Lemma 4.4 and Lemma 4.6, then solution is permanent.

Example 4.7. T = [2k, 2k + 1], k [member of] N k start with 0.

[x.sup.[DELTA]](t) = (2 - 1/2t + 2) - exp(x) - 0.5 exp(y)/exp(x) + exp(y), [y.sup.[DELTA]](t) = -1 + (3 + 1/t+1) exp(x)/exp(x) + exp(y).

Example 4.7 satisfies all conditions of Lemma 4.4 and Lemma 4.6, therefore, the solution is permanent.

5. Global Attractivity

Definition 5.1. A positive solution (x * (t),y * (t)) of (4.1) is said to be globally attractive if any other positive solution (x(t), y(t)) of (4.1) satifies [lim.sub.t[right arrow][infinity]] [absolute value of (x*(t) - x(t))] = 0, [lim.sub.t[right arrow][infinity]] [absolute value of (y*(t) - y(t))] = 0.

Theorem 5.2. In addition to conditions of Lemma 4.4 and Lemma 4.6 if [a.sub.1], [a.sub.2] [member of] (0, 1), [delta] > 0 and

[mathematical expression not reproducible],

then system (4.1) is globally attractive.

Proof. For any positive solutions ([x.sub.1](t), [y.sub.1](t)) and ([x.sub.2](t), [y.sub.2](t)) of system (3.1), it follows from Lemma 4.4 and Lemma 4.6 that lim [inf.sub.t[right arrow][infinity]] [x.sub.i](t) < [g.sub.1], lim [sup.sub.t[right arrow][infinity]], [x.sub.i](t) < [G.sub.1], lim [inf.sub.t[right arrow][infinity]](t) < [g.sub.2] and lim [sup.sub.t[right arrow][infinity]], [x.sub.i](t) < [G.sub.2] for i = 1, 2.

Let [V.sub.1](t) = [absolute value of (ln [x.sub.1](t) - ln [x.sub.2](t))], A = [alpha] + [beta][x.sub.1](t) + m(t)[y.sub.1](t), B = [alpha] + [beta][x.sub.2](t) + m(t)[y.sub.2](t). If t is right dense,

[mathematical expression not reproducible].

By using mean value theorem, we have

(5.1) [x.sub.1](t) - [x.sub.2](t) = exp(ln [x.sub.1](t)) - exp(ln [x.sub.2](t)) = [xi](t)(ln [x.sub.1](t) - ln [x.sub.2](t))

where [xi](t) is between [x.sub.1](t) and [x.sub.2](t). If t is right scattered, using Young's inequality and (5.1), we obtain,

[mathematical expression not reproducible].

Therefore,

[mathematical expression not reproducible].

Let [V.sub.2](t) = [absolute value of (ln [y.sub.1](t) - ln [y.sub.2](t))]. By using mean value theorem, we get

(5.2) [y.sub.1](t) - [y.sub.2](t) = exp(ln [y.sub.1](t)) - exp(ln [y.sub.2](t)) = [[xi].sub.2](t)(ln [y.sub.1](t) - ln [y.sub.2](t))

where [[xi].sub.2](t) lies between [y.sub.1](t) and [y.sup.2](t). If t is a right scattered point, by (5.2) we have

[mathematical expression not reproducible].

If t is right dense, then

[mathematical expression not reproducible]

Thus

[mathematical expression not reproducible].

Let us define a Lyapunov function as V(t) := [a.sub.1][V.sub.1](t) + [a.sub.2][V.sub.2](t), [a.sub.1], [a.sub.2] [member of] (0, 1). [V.sup.[DELTA]](t) = [a.sub.1][V.sup.[DELTA].sub.1](t) + [a.sub.2][V.sup.[DELTA].sub.2](t).

[mathematical expression not reproducible].

By assumption

[V.sup.[DELTA]](t) < -[delta][[absolute value of ([x.sub.1](t) - [x.sub.2](t))] + [absolute value of ([y.sub.2](t) - [y.sub.1](t))]].

Integrating both sides of the above inequality from [t.sub.1] to t, we get

[mathematical expression not reproducible].

Then,

[mathematical expression not reproducible],

[lim.sub.t[right arrow][infinity]] [[absolute value of ([x.sub.1](t) - [x.sub.2](t))]] = 0 and [lim.sub.t[right arrow][infinity]][[absolute value of ([y.sub.1](t) - [y.sub.2](t))]] = 0.

Hence, we get the desired result.

Corollary 5.3. In addition to conditions of Lemma 4.4 and Lemma 4.6 if [a.sub.1], [a.sub.2] [member of] (0, 1), [alpha](t) = 0, [delta] > 0 and

[mathematical expression not reproducible],

then, system (4.1) is globally attractive.

Example 5.4. T = [2k, 2k + 1], k [member of] N k start with 0.

[x.sup.[DELTA]](t) = (0.5 - 0.1/t + 1) - exp(x) - 0.01 exp(y)/exp(x) + exp(y), [y.sup.[DELTA]](t) = -0.1 + 0.2 exp(x)/exp(x) + exp(y).

Example 5.4 satisfies Corollary 5.3, therefore solution of this system is globally attractive.

Although we take several different initial conditions, the solutions (exp(x(t)), exp(y(t))) approaches to 0.5 in each case. Therefore numeric solution of Example 5.4 shows the global attractivity.

Received April 11, 2016

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NESLIHAN NESLIYE PELEN, AYSE FEZA GUVENILIR, AND BILLUR KAYMAKCALAN

Ondokuz Mays University, Faculty of Science, Department of Mathematic Samsun, TURKEY

Ankara University, Faculty of Science, Department of Mathematics Ankara, TURKEY

Cankaya University, Department of Mathematics and Computer Science 06810, Ankara, TURKEY

Corresponding author: Neslihan Nesliye Pelen

Caption: FIGURE 1. Numeric solutions of Example 4.7 show the permanence.

Caption: FIGURE 2. Initial conditions in this example are x(0) = 3, y(0) = 2.

Caption: FIGURE 3. Initial conditions in this example are x(0) = 0, y(0) = 0.

Caption: FIGURE 4. Initial conditions in this example are x(0) = 8, y(0) = 1.

Caption: FIGURE 5. Initial conditions in this example are x(0) = 10, y(0) = 10.
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Author:Pelen, Neslihan Nesliye; Guvenilir, Ayse Feza; Kaymakcalan, Billur
Publication:Dynamic Systems and Applications
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Date:Mar 1, 2017
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