Complete mathematical analysis of predator–prey models with linear prey growth and...

Applied Mathematics and Computation 162 (2005) 523–538

www.elsevier.com/locate/amc

Complete mathematical analysis ofpredator–prey models with linear preygrowth and Beddington–DeAngelis

functional response

Dobromir T. Dimitrov, Hristo V. Kojouharov *,1

Department of Mathematics, University of Texas at Arlington, P.O. Box 19408,

Arlington, TX 76019-0408, USA

Abstract

The dynamics of a predator–prey model with a Beddington–DeAngelis functional

response and linear intrinsic growth rate of the prey population is fully analyzed.

Conditions on local and global stability of the interior equilibrium are established. The

equilibria type are determined. All possible global asymptotic behaviors of the system

are considered, including the determination of the extinction conditions and existence of

periodic orbits. It is shown that mutual interference between predators can alone sta-

bilize predator–prey interactions even when only a linear intrinsic growth rate of the

prey population is considered in the mathematical model. Additional biological impli-

cations and a set of numerical simulations supporting the analysis are also presented.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Predator–prey; Beddington–DeAngelis functional response; Poincare–Bendixson The-

orem; Dulac�s criterion

* Corresponding author.

E-mail addresses: [email protected] (D.T. Dimitrov), [email protected] (H.V. Kojouharov).1 Kojouharov was supported in part by NSF Grant DMS 0107439 and UTA Grant REP

14748717.

0096-3003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2003.12.106

mail to: [email protected]

524 D.T. Dimitrov, H.V. Kojouharov / Appl. Math. Comput. 162 (2005) 523–538

1. Introduction

The dynamic interaction between predators and their prey has long been one

of the dominant themes in mathematical biology due to its universal existence

and importance. The popular method of modeling that interaction consists of a

system of two differential equations with a simple correspondence between the

prey consumption and the predator production. A crucial element of all models

is the so-called ‘‘functional response’’ or ‘‘feeding rate’’, which is the function

representing the prey consumption per unit time.

The earliest expression of the feeding rate F ¼ fxy is directly proportional tothe product of the predator y and prey x densities (Lotka [12], Volterra [14]).These models have fundamental importance for the predator–prey theory. How-

ever, the formulation appears inadequate under some conditions of prey abundance.

One of the most popular functional responses F ¼ fxyxþc is known as the

Michaelis–Menten–Holling type (May [13]). This expression extends the range

of values of x and y over which the feeding term is realistic. However, in somesituations the increase of the feeding rate is not proportional to the increase of

the predator density, as a result of mutual interference between predators,which decreases the efficiency of predation.

The ratio-dependent functional response F ¼ fxyxþby incorporates mutual

interference by predators. It was analyzed by Berezovskaya et al. [4], Kuang

and Beretta [11] and Hsu et al. [9] among others. It produces richer dynamics

than the previous expressions, but it is sometimes criticized, because of its

behavior at low densities. Normally one would expect that the predator growth

rate be negative when both populations fall below some critical level, because

the food-searching effort becomes really big. The ratio-dependent model sug-gests the opposite in some cases.

The Beddington–DeAngelis functional response F ¼ fxybþwxþy was introduced

by DeAngelis [7] as a solution of the observed problems in the classic predator–

prey theory (with Michaelis–Menten–Holling type functional response).

Independently, Beddington [3] offered the same form of a functional response

for describing parasite–host interactions. Later, a similar idea was applied to a

nutrient-producer–consumer modeling by Antonios and Hallam [2] and by

Antonios [1]. The Beddington–DeAngelis functional response has an extraterm in the denominator which models mutual interference between predators.

It represents most of the qualitative features of the ratio-dependent models, but

avoids the ‘‘low densities problem’’, which is usually the source of controversy.

The original predator–prey model, introduced by DeAngelis [7] has the form

x0 ¼ ax� fxybþ wxþ y

� g1x2;

y0 ¼ efxybþ wxþ y

� dy � g2y2;ð1Þ

D.T. Dimitrov, H.V. Kojouharov / Appl. Math. Comput. 162 (2005) 523–538 525

where all parameters are positive. The functions xðtÞ and yðtÞ are the densitiesof the prey and the predator, respectively, a is the intrinsic growth rate of theprey, d is the mortality rate of the predator, f is the feeding parameter, e is theconversion efficiency parameter, gi are the intraspecies interference parameters,w is a food weighting factor, that correlates inversely with the prey density atwhich feeding saturation occurs and b is a normalization coefficient that relatesthe densities of the predator and prey to the environment in which they

interact.

The corresponding model with logistic-growth rate of the prey population

(g1 > 0, g2 ¼ 0) was partially analyzed by Cantrell and Cosner [6] and Tzy-WeiHwang [10]. However the problem for the uniqueness of a limit cycle remains

open. In the present article we give a complete mathematical analysis of the

dynamics of the predator–prey model with Beddington–DeAngelis functional

response (1) assuming the interspecies density-dependent effects as negligible

ðg1 ¼ g2 ¼ 0Þ. Biologically, such systems with linear intrinsic growth rates ofthe prey populations are relevant in scenarios when the prey populations are

continuously suppressed far below their carrying capacity. The dissipativity of

the ‘‘logistic’’ system (g1 > 0, g2 ¼ 0) allows the direct use of the Poincare–Bendixson Theorem (Theorem 12.1 [8]), which is not applicable in the analysis

of the ‘‘linear’’ system ðg1 ¼ g2 ¼ 0Þ.After a change of variables u ¼ wx

b ; v ¼yb ;A ¼ f

a ;D ¼ da and E ¼ ef

wa

� �and an

appropriate time rescaling ðt ! at) the following nondimensional ‘‘linear’’model yields:

u0 ¼ u� Auv1þ uþ v

;

v0 ¼ Euv1þ uþ v

� Dv:ð2Þ

The paper is organized as follows. In Section 2 we state our main analytical

results on equilibria and global dynamics of the above system. Section 3

contains proofs of the main theorems. In the last section, biological implica-

tions, numerical simulations and future research directions are outlined.

2. Main results

The problem described by system (2) is well posed, because the u and v axesare invariant under the flow of the system. A solution starting in the interior of

the first quadrant never leaves it. This mathematical fact is consistent with thebiological interpretation of the system. In order to analyze the behavior of the

system we need to establish the existence and stability of the equilibria and then

consider the global dynamics of System (2).


2.1. Equilibria analysis

The system always has the trivial equilibrium ð0; 0Þ. To establish the exis-tence condition for the interior, nontrivial, equilibrium we need to consider the

isoclines of the system:

Fig. 1

and wh

first qu

v ¼ uA� 1þ

1

A� 1 ; u-isocline;

v ¼ ðE � DÞuD

� 1; v-isocline:

They intersect in the interior of the first quadrant if and only if

AE � E � AD > 0: ð3Þ

Under the above condition the interior equilibrium can be written as

ðu�; v�Þ ¼ ADAE � E � AD

;E

AE � E � AD

� �:

The isoclines divide the first quadrant into four regions with trajectory

directions given in Fig. 1. The Jacobian of the system (2) is given by

Jðu; vÞ ¼1� Avð1þvÞ

ð1þuþvÞ2 � Auð1þuÞð1þuþvÞ2

Evð1þvÞð1þuþvÞ2

Euð1þuÞð1þuþvÞ2 � D

0@ 1A: ð4Þ

The trivial equilibrium ð0; 0Þ is always locally asymptotically unstable. It is asaddle point whose stable manifold is the v-axis. For the interior equilibriumðu�; v�Þ the following propositions hold.

. Isoclines of system (2) with trajectory directions, when the interior equilibrium exists (left)

en the interior equilibrium does not exist, but both isoclines still intersect the interior of the

adrant (right).


Proposition 1. If A < E then the interior equilibrium ðu�; v�Þ is locally asymp-totically stable. If A > E then ðu�; v�Þ is locally asymptotically unstable.

Note 1. If A ¼ E then the linearization suggests that the interior equilibrium isa center or a spiral point (can be stable or unstable). This case is the object of

our further investigations in Theorem 4.

Proposition 2. Let the interior equilibrium ðu�; v�Þ exists and A 6¼ E.

(a) If DP 4AðA� 1Þ then ðu�; v�Þ is a stable node.(b) If A� 1 < D < 4AðA� 1Þ then ðu�; v�Þ is a stable node for E < bE and a sta-

ble spiral point for E > bE.(c) If D ¼ A� 1 then ðu�; v�Þ is a stable spiral point.(d) If D < A� 1 then ðu�; v�Þ is an unstable node for E < bE, an unstable spiral

point for bE < E < A and a stable spiral point for E > A.

Here bE ¼ ADð2A�1Þþ2AffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiADðA�1ÞðDþ1Þ

p4AðA�1Þ�D .

2.2. Global dynamics analysis

Here we concentrate our efforts on the existence of the periodic orbits and

the asymptotic behavior of the solutions of system (2).

Lemma 1. If the interior equilibrium exists and A 6¼ E then there is no periodicorbit of system (2).

Theorem 1. If AE � E � AD6 0 then all trajectories are unbounded. E > Dyields that limt!1 uðtÞ ¼ 1 and limt!1 vðtÞ ¼ 1 in the region 1 (Fig. 1(right)).E6D yields that limt!1 uðtÞ ¼ 1 and limt!1 vðtÞ ¼ 0.

Theorem 2. If AE � E � AD > 0 and A < E then the interior equilibrium ðu�; v�Þis not only locally asymptotically stable (l.a.s.), but also globally asymptoticallystable (g.a.s.), i.e., all trajectories approach ðu�; v�Þ, as t ! 1.

Theorem 3. If AE � E � AD > 0 and A > E then the interior equilibrium ðu�; v�Þis unstable. If, in addition, EPDþ 1 then every trajectory circles indefinitely andcounterclockwise around ðu�; v�Þ moving away from it. If E < Dþ 1 then everytrajectory stabilizes in the region 1 (Fig. 1(left)), where limt!1 uðtÞ ¼limt!1 vðtÞ ¼ 1.

Theorem 4. If AE � E � AD > 0 and A ¼ E then the interior equilibrium ðu�; v�Þis a global center, which means that all trajectories (except ðu�; v�Þ) are periodicorbits containing ðu�; v�Þ in their interior.


3. Proofs of the main results

Proof (Proposition 1). After some calculations, we find the following expres-

sion for the determinant of the Jacobian (4): detðJðu�; v�ÞÞ ¼ DAv� > 0. This

implies that the eigenvalues of Jðu�; v�Þ are both real with equal signs orcomplex conjugates.

For the trace of the Jacobian (4) we have that TrðJðu�; v�ÞÞ ¼ ðA�EÞDAE .

Therefore, A > E yields that the eigenvalues are positive real numbers or

complex conjugates with positive real parts. Thus the interior equilibrium is

asymptotically unstable. Similarly if A < E then the equilibrium is asymptoti-cally stable. h

Proof (Proposition 2). The type of the interior equilibrium ðu�; v�Þ in the caseA 6¼ E depends on the sign of

Dðu�; v�Þ ¼ Tr2ðJðu�; v�ÞÞ � 4 detðJðu�; v�ÞÞ

¼ DðA� EÞ2 � 4AEðAE � E � ADÞA2E2

¼ ðD� 4AðA� 1ÞÞE2 þ 2ADð2A� 1ÞE þ DA2

A2E2:

If Dðu�; v�ÞP 0 then the interior equilibrium is a node, otherwise it is a spiral

point (by Theorem 4.4 [5]). The existence condition (3) of ðu�; v�Þ yields A > 1and E > AD

A�1.

Let dðxÞ ¼ ax2 þ bxþ c, where a ¼ D� 4AðA� 1Þ, b ¼ 2ADð2A� 1Þ andc ¼ DA2. Obviously b > 0 and c > 0. The discriminant b2 � 4ac ¼ 16A3ðA� 1ÞðDþ 1Þ > 0. Thus dðxÞ ¼ 0 has real roots x1=2 ¼

�ADð2A�1Þ�2AffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiADðA�1ÞðDþ1Þ

pD�4AðA�1Þ .

We now consider the four cases in Proposition 2:

(a) If DP 4AðA� 1Þ (region 1, Fig. 2(left)) then aP 0 and both roots x1 and x2are negative (in the case a ¼ 0 there is only one root and it is negative).Since A < AD

A�1 < E then ðu�; v�Þ is a stable node.In the cases (b), (c) and (d) we have D < 4AðA� 1Þ. Then a < 0 andx1x2 < 0. Therefore dðxÞ ¼ 0 has a unique positive root and it is equal to

bE ¼ ADð2A� 1Þ þ 2AffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiADðA� 1ÞðDþ 1Þ

p4AðA� 1Þ � D

:

In addition d ADA�1� �

> 0. Thus ADA�1 <

bE.(b) If D < 4AðA� 1Þ and D > A� 1 (region 2, Fig. 2(left)) then A < AD

A�1 <bE,

which implies that ðu�; v�Þ is a stable node if E < bE and a stable spiral pointif E > bE.

(c) If D < 4AðA� 1Þ and D ¼ A� 1 then A ¼ ADA�1 ¼ bE, which implies that

ðu�; v�Þ is a stable spiral point.

Fig. 2. Regions of the equilibrium type determination in the plane ðA;DÞ (left) and a graph of atrajectory, that makes one circle around ðu�; v�Þ (right).


(d) If D < 4AðA� 1Þ and D < A� 1 (region 3, Fig. 2(left)) then dðAÞ < 0.Therefore AD

A�1 <bE < A. In this case ðu�; v�Þ is an unstable node if E < bE,

an unstable spiral point if bE < E < A and a stable spiral point if E > A. h

In what follows, R2þ represents the region R2þ ¼ fðu; vÞ : u > 0; v > 0g.

Proof (Lemma 1). Let us define

Lðu; vÞ ¼ o

ouðbðu; vÞF ðu; vÞÞ þ o

ovðbðu; vÞGðu; vÞÞ; ð5Þ

where F ðu; vÞ ¼ u� Auv1þuþv, Gðu; vÞ ¼ Euv

1þuþv � Dv and bðu; vÞ ¼ 1uv. This implies

that Lðu; vÞ ¼ A�Eð1þuþvÞ2. If A > E then Lðu; vÞ > 0 in R2þ and if A < E then

Lðu; vÞ < 0 in R2þ. Therefore, there is no periodic orbit of System (2) by Dulac�scriterion (Theorem 4.8 [5]). h

The proofs of our main Theorems 1–4 are based on the following lemmas.

Lemma 2. Let us consider a trajectory, which starts at ðu�; v0Þ, where v0 < v�,makes a circle around ðu�; v�Þ and intersects the line u ¼ u� at ðu�; v1Þ at time t,where v1 < v� (Fig. 2(right)). If A < E (¼ or >), then v1 > v0 (¼ or <)respectively.

Proof. We will apply the Green�s Theorem
Z ZK
oQou

�� oP

ov

�dA ¼

ZcðP duþ QdvÞ ð6Þ

to the functions P ðu; vÞ ¼ �bðu; vÞGðu; vÞ and Qðu; vÞ ¼ bðu; vÞF ðu; vÞ, where b,F and G are defined as in (5), c is the closed curve containing the part of thetrajectory c1 from ðu�; v0Þ to ðu�; v1Þ and the line segment c2 from ðu�; v1Þ to


ðu�; v0Þ, and K is the region inside c. Similarly to the proof of Lemma 1 we canshow that the left-hand side of (6) is equal to

R RK

A�Eð1þuþvÞ2 dA.

On the other hand, the right-hand side of (6) is equal to
ZcðP duþ QdvÞ ¼
Zc1

ðP duþ QdvÞ þZc2

ðP duþ QdvÞ;

where
Zc1
ðP duþ QdvÞ ¼Z t

0

�v0

uvdu

�þ u0

uvdv�

¼ 0

and
Zc2
ðP duþ QdvÞ ¼Z v0

v1

1

v

�� A1þ u� þ v

�dv ¼ ln v0ð1þ u� þ v1ÞA

v1ð1þ u� þ v0ÞA

!:

Let us analyze the function aðxÞ ¼ 1þ xb

� �A � 1� xc, where b ¼ 1þ u� þ v0,

c ¼ v0 and x 2 ð�v0; v� � v0Þ. If A < E, then the left-hand side of (6) is less than

0. Therefore ln v0ð1þu�þv1ÞA

v1ð1þu�þv0ÞA

� �< 0, which is equivalent to aðv1 � v0Þ < 0. Simi-

larly, A ¼ E implies that aðv1 � v0Þ ¼ 0 and A > E implies that aðv1 � v0Þ > 0.In order to prove the lemma we need to determine the sign of aðxÞ in

ð�v0; v� � v0Þ. The derivative of aðxÞ is a0ðxÞ ¼ Ab 1þ x

b

� �A�1 � 1c. Since A > 1 and

bAc ¼

ðA�1Þv�þv0Av0

> 1 then a0ð0Þ ¼ Ab � 1

c < 0 and a0ðxÞ ¼ 0 has only one solutionx0 > 0. Therefore aðxÞ decreases for x < x0 and increases for x > x0 aslimt!1 aðxÞ ¼ 1. Since að0Þ ¼ 0 then aðxÞ > 0 for x < 0 and aðxÞ < 0 forx 2 ð0; x1Þ, where x1 satisfies x1 > x0. To finish the proof it is sufficient to showthat x1 > v� � v0, which is equivalent to aðv� � v0Þ < 0. Simple calculationsyield

aðv� � v0Þ ¼ð1þ u� þ v�ÞA

ð1þ u� þ v0ÞA� v�

v0¼ ðAv�ÞA

ððA� 1Þv� þ v0ÞA� v�

v0:

The last expression is negative, if and only if

ðv�ÞA�1v0 <A� 1A

v��

þ v0A

�A

;

which is true by the generalized Jensen inequality. h

Lemma 3. If ðuðtÞ; vðtÞÞ is a solution of the system (2) and uðtÞ is bounded thenvðtÞ is bounded.


Proof. Let uðtÞ < u0 for any t. Then v0 < Eu0 � Dv. A standard comparisonargument shows that limt!þ1 sup vðtÞ6 Eu0

D . h

Lemma 4. Let the interior equilibrium ðu�; v�Þ exist and ðuðtÞ; vðtÞÞ be an un-bounded solution of the system (2), which remains in the region 1 (Fig. 1(left)) fort > t0. Then limt!1 vðtÞ ¼ 1.

Proof. Since v < uþ1A�1 in the region 1 (Fig. 1(left)) then

v0 >ðA� 1ÞEuAð1þ uÞ

�� D

�v:

Let d ¼ ðA�1ÞEA � D. The existence condition (3) of ðu�; v�Þ yields d > 0. Thus

there exists u0 such thatðA�1ÞEuAð1þuÞ � D > d

2for u > u0. If the trajectory is un-

bounded and remains for t > t0 in the region 1, where u0 > 0 and v0 > 0, thenthere exists t1 > t0 such that uðt1Þ > u0 (by Lemma 3). Therefore vðtÞ growsexponentially for t > t1. h

Lemma 5. Let the interior equilibrium ðu�; v�Þ exist. If A6E or Dþ 16E < A,then every trajectory starting in the region 1 (Fig. 1(left)) intersects the u-isoclineand enters in the region 2 (Fig. 1(left)).

Proof. The proof is based on the estimate of the ratio vu. Assume that a tra-

jectory does not leave the region 1. Then it stays below the u-isocline and vu is

bounded from above. On the other hand

vu

� �0¼ Euþ Av

1þ uþ v

�� 1� D

�vu:

If we prove that

F ðu; vÞ ¼ Euþ Av1þ uþ v

� 1� D > a ð7Þ

for some a > 0 and for t > t0 then that will imply limt!1vu ¼ 1, which is a

contradiction with our assumption. Before proving (7) we observe the fol-lowing:

(1) oFov ¼

ðA�EÞuþA

ð1þuþvÞ2 is positive if APE and negative if A < E and u is sufficiently

large ðu > u1 ¼ AE�AÞ.

(2) F ðu; vÞ > 0 is equivalent to ðA� 1� DÞv > ð1þ D� EÞuþ ð1þ DÞ.(3) F ðu�; v�Þ ¼ 0, therefore ðu�; v�Þ lays on the line ðA� 1� DÞv ¼

ð1þ D� EÞuþ ð1þ DÞ.

Let us define the region Rw ¼ ðu; vÞ : u > w; v < uþ1A�1

� �for any wP u�.

Obviously Rw is a part of the region 1 for every wP u�.

Fig. 3. Sketch of the regions used in the proof of Lemma 5 (left) and Theorem 3 (right).


To prove Lemma 5 in the case of A6E we consider the following threesubcases:

(1) Let A < E and A6Dþ 1. Then E > ADA�1 PDþ 1. Thus F ðu; vÞ > 0 is equiv-

alent to v < E�D�1Dþ1�A u� 1þD

Dþ1�A if A < Dþ 1 (or u > u� if Aþ D). This impliesthat F ðu; vÞ > 0 in Ru� (Fig. 3(left)). Moreover

oFov < 0 in Ru1 . Therefore

infRu1

F ðu; vÞ ¼ infuF u;

uþ 1A� 1

� �¼ inf

u

EðA� 1ÞA

u1þ u

�� D

�:

Since d ¼ EðA�1ÞA � D > 0 (by the existence condition (3) of ðu�; v�Þ) then

there exists u2 > u1 such thatEðA�1Þ

Au1þu � D > d

2in Ru2 . By Lemma 3 there

exists a time t0 such that uðt0Þ > u2, which guarantees that F ðu; vÞ > d2for

t > t0.(2) Let A < E and A > Dþ 1. Then F ðu; vÞ > 0 is equivalent to

v > � E�D�1A�D�1 uþ 1þD

A�1�D. If u0 ¼ 1þDE�D�1 then F ðu; vÞ > 0 in Ru0 . Using the same

arguments (and the same d) as in the previous subcase we can find t0 suchthat F ðu; vÞ > d

2for t > t0.

(3) Let A ¼ E. Then A ¼ E > Dþ 1 (by the existence condition (3) of ðu�; v�Þ)and F ðu; vÞ > 0 is equivalent to v > � E�D�1

A�D�1 uþ 1þDA�1�D. If u0 ¼ 1þD

E�D�1 thenF ðu; vÞ > 0 in Ru0 . Moreover

oFov > 0 in Ru0 . Thus infRu0

F ðu; vÞ ¼infu F ðu; 0Þ ¼ infu Eu

1þu � 1� D� �

.

Since d ¼ E � D� 1 > 0 then there exists u1 > u0 such that Eu1þu � 1� D > d

2

in Ru1 . Lemma 3 implies that there exists a time t0 such that uðt0Þ > u1,which guarantees that F ðu; vÞ > d

2for t > t0.

To prove Lemma 5 in the case of Dþ 16E < A we consider the followingtwo subcases:

(1) Let Dþ 1 < E < A. This case is identical to the subcase 3 of the case A6Eand using the same arguments we can find t0 such that F ðu; vÞ > d

2for t > t0.


(2) Let Dþ 1 ¼ E < A. Then F ðu; vÞ > 0 is equivalent to v > 1þDA�D�1. Therefore

F ðu; vÞ > 0 in R1 ¼ ðu; vÞ : 1þDA�D�1 < v < uþ1

A�1� �

. Lemmas 3 and 4 imply thatthe trajectory enters the region R1 and remains there. Using the samearguments as in the previous cases we can find t0 such that F ðu; vÞ > d

2

for t > t0.

Proof (Theorem 1). Since AE � E � AD6 0 then the only equilibrium is the

trivial equilibrium ð0; 0Þ, which is a saddle point, with a stable manifold the v-axis. Therefore there are no trajectories approaching ð0; 0Þ. By the Poincare–Bendixson Theorem (Theorem 12.1 [8]) all trajectories are unbounded. In the

case where E > D the v-isocline intersects the first quadrant (Fig. 1(right)). Alltrajectories enter in the region 1 and limt!1 uðtÞ ¼ limt!1 vðtÞ ¼ 1 (by Lem-

mas 3 and 4). In the case when E6D the region 1 does not exist. All trajectoriesenter in the region 2, where limt!1 uðtÞ ¼ 1 and limt!1 vðtÞ ¼ 0. h

Proof (Theorem 2). The interior equilibrium ðu�; v�Þ is the only l.a.s. equilib-rium and there is no periodic orbit. By the Poincare–Bendixson Theorem anytrajectory approaches ðu�; v�Þ or it is unbounded as t ! 1. Every trajectorystarting in regions 2, 3 or 4 (Fig. 1(left)) approaches ðu�; v�Þ or enters the region1 and intersects the line u ¼ u� at ðu�; v0Þ, where v0 < v�. By Lemma 5 itintersects the u-isocline and enters in the region 2. Repeating the above argu-ments the trajectory approaches ðu�; v�Þ or intersects u ¼ u� again at ðu�; v1Þ,where v1 < v� (in the region 1). By Lemma 2 we know that v0 < v1. Then thetrajectory enters the region K (Fig. 2(right)) and remains in it. Therefore by the

Poincare–Bendixson Theorem the trajectory approaches ðu�; v�Þ as t ! 1. h

Proof (Theorem 3). Using the results from Section 2.1, the interior equilibrium

ðu�; v�Þ is unstable under the above conditions. By the Poincare–BendixsonTheorem all trajectories are unbounded. By Lemma 2 every trajectory circlescounterclockwise around the interior equilibrium and moves away from it

following the path: region 2fi region 3fi region 4fi region 1 (Fig. 1(left)). Thequestion here is whether the trajectory remains in the region 1 or goes back to

region 2 for a new circle. In the case A > EPDþ 1 the trajectory always re-turns to the region 2 (by Lemma 5).

Let E < Dþ 1. Then A > E > ADA�1 yields A > Dþ 1 and 1þD

A�1�D > 1A�1. If

F ðu; vÞ is defined as in Lemma 5 then F ðu; vÞ > 0 is equivalent to

v > 1þD�EA�1�D uþ 1þD

A�1�D. Therefore F ðu; vÞ < 0 and vu

� �0< 0 in the region

R1 ¼ ðu; vÞ : u > DE�D ; v <

1þD�EA�1�D u

� �(Fig. 3(right)). Any trajectory starting in R1

remains in it, because the ratio vu decreases. Also limt!1 uðtÞ ¼ limt!1 vðtÞ ¼ 1

(by Lemma 3).


The next step is to prove that every trajectory enters the region R1. Let usconsider the system (2) with the following time rescaling (t ! �Dt). The newsystem has the form

Fig. 4

(right)

u0 ¼eEuv

1þ uþ v� eDu;

v0 ¼ v�eAuv

1þ uþ v:

ð8Þ

System (8) is identical to system (2), with interchanged u and v and parameterseA ¼ ED,eE ¼ A

D andeD ¼ 1

D, instead of the old A, E and D, respectively. System (8)has the same interior equilibrium which exists under the same existence con-

dition. Condition A > E is equivalent to eA < eE and by Theorem 2 the interiorequilibrium ðu�; v�Þ is a g.a.s. equilibrium for the system (8). Therefore a tra-jectory of the system (2) starting in the region R1 approaches the interiorequilibrium ðu�; v�Þ as t ! 1, which means that every trajectory of the system(2) that enters the region R1 approaches ðu�; v�Þ as t ! �1. Let ðUðtÞ; V ðtÞÞ bean ‘‘extended’’ trajectory defined for t 2 ð�1;1Þ, with limt!�1ðUðtÞ; V ðtÞÞ ¼ðu�; v�Þ and ðUðtÞ; V ðtÞÞ 2 R1 for t > t0. Then any trajectory that circles indef-initely around ðu�; v�Þ and moves away from it will definitely intersect

ðUðtÞ; V ðtÞÞ, which is impossible. h

Proof (Theorem 4). Let us first consider the trajectory with starting point

ðu�; v0Þ, where v0 < v�. By Lemma 5 it intersects the u-isocline and enters inthe region 2 (Fig. 1(left)). After that it approaches ðu�; v�Þ in the region 3 or theregion 1 (Fig. 4) or intersects u ¼ u� again at ðu�; v1Þ, where v1 < v�. In thesecond case we know that v0 ¼ v1 by Lemma 2, which means that it is aperiodic orbit. To exclude the possibility of the trajectory approaching ðu�; v�Þin the region 3 or the region 1 we follow the proof of Lemma 2 for the same

. Possible graph of a trajectory that tends to ðu�; v�Þ in the region 1 (left) or the region 3

.


function F ðu; vÞ, but over the region K1 (Fig. 4), closed by the curve c1, whichconsists of the part of the trajectory c11 from ðu�; v0Þ to ðu�; v�Þ and the linesegment c12 from ðu�; v�Þ to ðu�; v0Þ. This implies that

0 ¼Zc1ðP duþ QdvÞ ¼

Zc11

ðP duþ QdvÞ þZc12

ðP duþ QdvÞ;

where
Zc11
ðP duþ QdvÞ ¼Z 1

0

�v0

uvdu

�þ u0

uv

�dvÞ ¼ 0

and

Zc12

ðP duþ QdvÞ ¼Z v0

v�

1

v

�� A1þ u� þ v

�dv ¼ ln v0ð1þ u� þ v�ÞA

v�ð1þ u� þ v0ÞA

!:

The last expression was analyzed in the proof of Lemma 5. It becomes zero

if and only if v� � v0 ¼ 0, which in this case is not true.So far we know that all trajectories which intersect the line segment

fðu�; vÞ : v 2 ð0; v�Þg are periodic orbits. It is not difficult to see (using Lemmas2 and 5) that if a trajectory does not intersect that line segment then it ap-

proaches ðu�; v�Þ as t ! 1. This implies that it will intersect the periodic tra-jectories that start at ðu�; v0Þ, where v0 < v� is sufficiently close to v�, which isimpossible. h

4. Discussion

In the previous two sections we have presented a complete analysis of theasymptotic behavior of the solutions of a predator–prey model with the

Beddington–DeAngelis functional response and linear intrinsic growth rate

of the prey population. In this section we discuss the biological implications of

our mathematical results (Theorems 1–4) and present numerical verification of

the analysis predictions. All simulations are obtained by using the adaptive

MATLABATLAB solver ode45 applied to system (2).

Consider first the case AE � E � AD6 0 (Theorem 1). It is equivalent to the

case ef 6 dwþ ea in the original parameters. In this case the maximum growthrate ef of the predator is relatively small and that guarantees the prey�s sur-vival. The survival of the predator depends on the additional condition ef

w > d.It means that the predator�s growth rate, in an environment of prey abundance,


must be greater than the predator�s mortality rate. Otherwise the mortality ratewill prevail and the predator will die out. Representative numerical simulationsfor this case (Theorem 1) are given in Fig. 5(a) and (b).

Next consider the case of the greatest biological interest in system (2), when

AE � E � AD > 0 and A < E (Theorem 2). In general, linear prey growth is

relevant only if the prey population is continuously suppressed far below their

carrying capacity. In this case this is done by the predator, which is strong

enough to control the prey�s population size. Theorem 2 demonstrates that thesuppression of prey can be stabilized by the mutual interference of the pre-

dators, without adding a more realistic logistic prey growth in the mathe-matical model. The stability condition A < E can be expressed in the originalparameters as w < e. The parameter w is an empirical weighting factor whichmeasures the abundance of the prey relative to the predator population and the

environment in which they interact. The increase of w causes the lowering ofthe saturation level of the predator. If w is relatively small then the predatorhas a chance to survive and both populations stabilize at the interior equilib-

rium. Representative numerical simulations for this case (Theorem 2) are given

in Fig. 5(c).The case AE � E � AD > 0 and A > E (Theorem 3) represents an envi-

ronment in which the predator is a real threat to the prey population. The

analysis shows that if the predator�s maximum growth rate ef is less thanwðd þ aÞ (the predator�s mortality rate plus the prey�s intrinsic growth ratemultiplied by the ‘‘food abundance’’ factor w) then the system will have a

chance to survive as both populations will grow unbounded after some initial

oscillations. It is possible during this initial oscillatory period one of the

population densities to fall below its ‘‘survival’’ minimum. That will cause theextinction of the predator or both species. On the other hand, if the predator

maximum growth rate is greater than wðd þ aÞ then the system will oscillateindefinitely and finally one of the populations will fall below its ‘‘survival’’

minimum. This will cause the extinction of the predator or both species.

Representative numerical simulations for this case (Theorem 3) are given in

Fig. 5(e) and (f).

The last case, AE � E � AD > 0 and A ¼ E (Theorem 4) has a significant

importance for the completeness of the mathematical analysis of the globalbehavior of the system. It shows that there exist a balance between the pred-

ator�s need for food and its saturation level and in this case is likely to expect aperiodic behavior as a long term prediction. This behavior is relatively

unstable. A small change in the parameters (caused by environmental changes,

for instance) forces the system to stabilize at the interior equilibrium (by

Theorem 2) or to oscillate indefinitely around the interior equilibrium moving

away from it, which results in the extinction of the predator or both species (by

Theorem 3). Representative numerical simulations for this case (Theorem 4)are given in Fig. 5(d).

Fig. 5. Numerical simulation supporting the theoretical results of Theorems 1–4. (a) A ¼ 3:0,D ¼ 1:0, E ¼ 1:4, u0 ¼ 1:0, v0 ¼ 1:0 (Theorem 1). (b) A ¼ 3:0, D ¼ 1:0, E ¼ 0:6, u0 ¼ 1:0, v0 ¼ 0:5(Theorem 1). (c) A ¼ 1:6, D ¼ 0:5, E ¼ 1:65, u0 ¼ u�, v0 ¼ v�

2(Theorem 2). (d) A ¼ 1:6, D ¼ 0:5,

E ¼ 1:6, u0 ¼ u�, v0 ¼ v�

2(Theorem 4). (e) A ¼ 1:6, D ¼ 0:5, E ¼ 1:55, u0 ¼ u�, v0 ¼ v�

2(Theorem 3).

(f) A ¼ 1:6, D ¼ 0:5, E ¼ 1:4, u0 ¼ u�

2, v0 ¼ v� (Theorem 3).



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