Dynamics of coral reef models in the presence of parrotfish

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1 Receive: 0 June 018 Accepte: 11 Octobe 018 DOI: /nm.10 Dynamics of coal eef moels in the pesence of paotfish Haniyeh Fattahpou 1 Hami R.Z. Zangeneh 1 Hao Wang 1 Depatment of Mathematical Sciences, Isfahan Univesity of Technology, Isfahan, Ian Depatment of Mathematical an Statistical Sciences, Univesity of Albeta, Emonton, Albeta, Canaa Coesponence Hami R.Z. Zangeneh, Depatment of Mathematical Sciences, Isfahan Univesity of Technology, Isfahan , Ian. hamiz@cc.iut.ac.i Abstact The ecline of coal eefs chaacteize by macoalgae incease has been a global theat. We consie a slightly moifie vesion of an oinay iffeential equation (ODE) moel popose in Blackwoo, Hastings, an Mumby [Theo. Ecol. 5 (01), pp ] that explicitly consies the ole of paotfish gazing on coal eef ynamics. We pefom complete stability, bifucation, an pesistence analysis fo this moel. If the fishing effot (f) is in between two citical values f 0 an f 1, then the system has a unique inteio equilibium, which is stable if f0 < f < f1 an unstable if f1 < f < f0.iff is less (moe) than these citical values, then the system has up to two (zeo) inteio equilibia. Also, we evelop a moe ealistic elay iffeential equation (DDE) moel to incopoate the time elay an teating it as the bifucation paamete, an we pove that Hopf bifucation about the inteio equilibia coul occu at citical time elays, which illustate the potential impotance of the inheent time elay in a coal eef ecosystem. Recommenations fo Resouce Manages One seious theat to coal eefs is ovefishing of gazing species, incluing high level of algal abunance. Fishing altes the entie ynamics of a eef (Hughes, Bai, & Bellwoo, 003), fo which the coal cove was peicte to ecline apily (Mumby, 006). One majo issue is to evese an evelop appopiate management to incease o maintain coal esilience. We have povie a etaile local an global analysis of moel (Blackwoo, Hastings, & Mumby, 01) an Natual Resouce Moeling. 018;e10. wileyonlinelibay.com/jounal/nm 018 Wiley Peioicals, Inc. 1of4

2 of4 Natual Resouce Moeling FATTAHPOUR ET AL. obtaine an ecologically meaningful attacting egion, fo which thee is a chance of stable coexistence of coal algal fish state. The healthy eefs switch to unhealthy state, an the macoalgae paotfish state becomes stable as the fishing effot inceases though some citical values. Also, fo some citical time elays, a switch between healthy an unhealthy eef states occus though a Hopf bifucation, which can only appea in the elay iffeential equation (DDE) moel. Eventually, fo lage enough time elay, oscillations appea an an unhealthy state occus. KEYWORDS coal eef, elay iffeential equation, Hopf bifucation, oinay iffeential equation, stability 1 INTRODUCTION Global ivesity an quality of life of millions of people living in topical coastal egions have been enangee by fequency an scale of human impacts on the health of wol scoaleefs (cf., Mumby, Hastings, & Ewas, 007; Panolfi, Babuy, & Sala, 003). The wolwie ecline of coal eefs, which is chaacteize by macoalgae incease, has emonstate an ugent nee to scale up the management effot base on impove unestaning of ecological ynamics that unelies the eef esilience (cf., Bellwoo, Hughes, Folke, & Nystom, 004). Mumby et al. (007) intouce a moel with gazing that shows a coal eef ecosystem that may lose esilience an shift to coal eplete state though euctions in gazing intensity, which has been analyze mathematically in Li, Wang, Zhang, an Hastings (014). This moel has been extene by Blackwoo et al. (01) in the explicit incopoation of paotfish abunance. It has been hypothesize that excessive fishing pessue is a potential cause of hysteesis (cf., Li et al., 014). Motivate by these papes, ou aim is to consie a moifie vesion of the oinay iffeential equation (ODE) moel in Blackwoo et al. (01) an to etemine explicitly the ole of paotfish abunance on gazing intensity an implement management on the coal eef system. The analysis of ecological systems often focuses on thei asymptotic behavio, but it is impotant to unestan the ole of tansient behavio. The inheent elay effect has been consiee to be impotant on coal algae paotfish inteactions (cf., Blackwoo & Hastings, 011), in which the authos iscusse the effect of time elays on the basis of attaction of stable equilibia, which in tun may have impotant management implications in coal eef as well as othe ecosystems exhibiting simila ynamical behavio. Algal tufs will not be bloome immeiately afte macoalgae ae gaze by paotfish. To account fo the time elay, an accoingly the time that affect the natual system, we evelop an analyze a moe ealistic elay iffeential equation (DDE) moel by teating the time elay as the bifucation paamete.

3 FATTAHPOUR ET AL. Natual Resouce Moeling 3of4 We consie the fishing effot an time elay as the bifucation paamete an analyze the local stability of equilibia fo the DDE moel. The pape is oganize as follows. In Section, we pesent ou basic assumptions an the ODE moel. In Section 3, we consie the existence, stability, an bifucation of all possible equilibia of the system an show the unifom pesistence esult. In Section 4, we constuct a DDE moel to account fo the ole of time elay in the coal eef ynamics. We pove that the Hopf bifucation can occu about the unique inteio equilibia. In Section 5, we pesent epesentative numeical simulations to suppot ou analytical esults. Finally, in Section 6, we conclue ou esults an povie biological intepetations. THE MODEL AND BASIC ASSUMPTIONS In this section, we consie the following moifie vesion of the ODE moel pesente in Blackwoo et al. (01): P P sp t ( βk ( C) ) M t C t T t g( P) M 1+ M+ T = amc + γmt, = TC C amc, g( P) M 1+ M+ T = γmt TC + C, = 1 fp. (1) (i) We assume that a paticula egion of the seabe is covee entiely by macoalgae ( M), coal ( C), an algal tufs ( T) so that M + C + T is kept constant, which by a escaling we may assume that M + C + T = 1. Theefoe, the faction of algal tufs is efine by T t M t T =1 M C an consequently = (ii) It is assume that coals ecuit an ovegow algal tufs at a ate, they have a natual motality ate of, ae ovegown by macoalgae at a ate a, an macoalgae spea vegetatively ove algal tufs at a ate γ. Space fee by coal motality is assume to be ecolonize by algal tufs. Aitionally, M/(1 + M + T) is simply the popotion of gazing that affects macoalgae. (iii) Changes in paotfish abunance, P, ae moele as logistic gowth with an intinsic ate of gowth s an a time vaying caying capacity such that β is the maximum caying capacity an 0 < K( C) 1 is a nonimensional tem that limits caying capacity of paotfish as a function of coal cove. We have simplifie the moel by the assuming that paotfish lacks aequate efugia, likely as a esult of huicane impacts esulting in a positive elationship between paotfish caying capacity an coal gowth. Thus, we use the caying capacity that is efine as K ( C) = 1 kc, whee 0 k < 1, so that thee is a shot tem positive esponse to coal ecuitment. Futhemoe, it is assume that motality esulting fom fishing effot is hel at a constant level f 0. (iv) Gazing intensity vaies epening on paotfish abunance so that the gazing is efine by as a function g( P). It is assume that the gazing intensity is popotional to paotfish abunance, elative to its maximum caying capacity, o g( P) = αp, whee α is a positive β C t.

4 4of4 Natual Resouce Moeling FATTAHPOUR ET AL. constant. It is natual to let α = g max, whee g max is the maximum possible gazing intensity an fo simplicity it is assume to be equal one. Thus, g( P) = P an this β fomulation guaantees that the gazing intensity will aive at its maximum only if fishing effot is eliminate ( f =0). Fo simplicity, we scale paotfish abunance an intouce the nonimensional vaiable P to be paotfish abunance elative to its maximum caying capacity, o P = P. Substituting this in the last equation of (1) leaves with β P t P = sp (1 ) fp, an the gazing function is now given by g( P) = P. Fo simplicity K( C) in notation, we set P =: P. Fom the afoementione an setting M x, C y, an P z, equations of the moel ynamics ae given by the following nonlinea system of ODEs: x t y t z t z = x γ γx + ( a γ) y xf( x, y, z), y = y[( ) ( + a) x y] yg( x, y, z), sz = z ( s f) zh( x, y, z). 1 ky () The bief esciption of the paamete selection an paamete values can be foun in Blackwoo et al. (01) an Mumby et al. (007), in which the authos liste the paamete values as follows: a = 0.1, = 0.44, s = 0.49, γ = 0.8, = 1, 0 f < 0.49, 0 k < 1. Hee, we only assume that all paametes ae positive an satisfy the following assumptions: s f a < γ < + a, >, 0 < < γ, 0 k <1. (3) s Fo convenience, we enote μ s f s. 3 EQUILIBRIA: STABILITY AND BIFURCATION In this section, we fin all possible equilibia an consie thei stabilities. By consieing the nullclines of the system (), we coul easily fin the following equilibia: The tivial state: O = (0, 0, 0). μ ( (1 k)+ k) The axial states: R = (0, 0, μ), N = (0,, 0), Q = (1, 0, 0). The bounay states: D = (0,, ), F = (1, 0, μ). Hee, we notice that a a + γ a + a aγ γ( a+ ) a + a aγ thee is anothe bounay state G =(,,0), which is not in the fist positive octant, so it is ignoe hee. μ γ

5 FATTAHPOUR ET AL. Natual Resouce Moeling 5of4 The inteio steay state: E* =( x*, y*, z* ) is the solution of F( x, y, z) = G ( x, y, z) = H( x, y, z) = 0, so that x* an z* satisfy x * y = z μ ky + a *, * = (1 *). (4) + a an y* is a solution of the following quaatic equation: whee K () y y + y +, (5) = a ( + a γ), = γ( a + )+ a( + a γ) μk( + a), = γ( a + ) μ( + a). This solution shoul belong to the inteval (0, ), because othewise x * <0. Notice that, une the assumption (3), the axial points (R, N, an Q) an the bounay points (D an F) always exist. Now, let us efine the following quantities: μ 0 a ( ) + ( + γ ) γ( a + ), μ. + k 1 (6) k + a ( )( + ) Notice that μ 0 is a function of k an μ0 = μ1 fo k = k 1 a a γ 0 γ ( a + ) 0 k 0 < k an 0 μ0 < μ1 fo 0 k < k 0. In the following, we fin conitions of inteio equilibium. Fist, we note that K(0) = ( + a)( μ μ), 1 ( ) ( ) k ( ) K =( + a) 1 ( μ μ), 0. Also, μ1 < μ0 fo whee μ 0 an μ 1 ae efine in (6). Let Y max be the maximum point of K ( y) an Δ = 4. We nee to consie the following cases sepaately: Case (I) 0 < μ 1 < μ 0 : Then, thee can be thee iffeent scenaios: (i) 0 < μ < μ 1 < μ 0 : In this case, K (0) > 0 an K ( ) > 0. Then, epening on the position of Y max, thee ae thee possibilities: (a) 0 Y : Then, K ()>max( y K(0), K( ))>0. max (b) Y max > : Then, K ( y) is inceasing in (0, ) an theefoe K ()> y K(0)>0. (c) Y max <0: Then, K ( y) is eceasing in (0, ) an theefoe K ()> y K( )>0. This implies that K ( y) is positive on (0, ) an theefoe thee is no inteio equilibium.

6 6of4 Natual Resouce Moeling FATTAHPOUR ET AL. (ii) 0 < μ1 < μ < μ0 : In this case, K (0) < 0 an K ( ) > 0, then K ( y) has at least one solution in (0, ). Depening on the position of Y max, thee ae two possibilities: (a) 0 < Y <. (b) Y max max. In eithe case, K ( y) has two positive solutions fom which only Δ the smalle solution, y * =,isin(0, ). (iii) 0 < μ1 < μ0 < μ: In this case, K (0) < 0 an K ( ) < 0. Again thee ae thee iffeent possibilities: (a) Ymax 0 o (b) Ymax. In these two cases, K ( y) is monotone on (0, ) an theefoe thee is no solution in the inteval of inteest. (c) 0 < Ymax < : Notice that this hols only if μ < μ < μ*, whee * a γ a a μ =: (3 + ) ( + ), * k( + a) μ* aγ ( + ) + ( + a)( a + γ). k ( + a) But μ* > μ > 0 always hols, as by hypothesis (3), a ( )[ γ+ + a] > 0. * Futhemoe, μ*, μ *, an μ 0 ae functions of paamete k. μ* an μ ae eceasing in * k, wheeas μ 0 is inceasing in k. Also, it can be shown that μ* (1) < μ1. Theefoe, epening on the value of Δ, K( y) can have up to two solutions in (0, ). Case (II) 0 < μ0 < μ1 : As in Case (I) an by simila agument, we can conclue the following: (i) μ < μ0 < μ1 : K ( y ) has no solution in inteval (0, ). (ii) 0 < μ0 < μ < μ1 : K ( y ) has a unique positive solution y = inteval (0, ). * + Δ (iii) μ0 < μ1 < μ: K ( y) can have up to two solutions in the inteval (0, ). We summaize the above iscussion in the following poposition. in the Poposition 1. (a) If 0 < μ1 < μ < μ0 o 0 < μ0 < μ < μ1, the system () has a unique inteio equilibium. (b) If μ < μ 1 < μ 0 o μ < μ0 < μ1, the system () has no inteio equilibia. (c) If μ1 < μ0 < μ o μ0 < μ1 < μ, the system () can have up to two inteio equilibia. Now, we stuy the local stability of the tivial state O, the axial states R, N, Q, an the bounay states D an F by calculating the eigenvalues of the Jacobian matix of the system ()

7 FATTAHPOUR ET AL. Natual Resouce Moeling 7of4 J ( xyz,, ) F + xfx xfy xfz = ygx G + ygy ygz zhx zhy H + zhz about iffeent equilibia. By looking at eigenvalues of J, we can easily check that O is an unstable noe, wheeas the axial solutions R, N, an Q ae always a sale. Now, we consie the bounay points: ( (1 k)+ k) Bounay point D. Eigenvalues ae λ1 = ( μ μ), λ = ( ) ( + ) 0, an λ3 = sμ. Une the assumption (3), λ an λ 3 ae always negative. If μ0 < μ, λ1 < 0, an D is a stable noe, if 0 < μ < μ0, λ1 > 0 an it is unstable. It can be easily checke that the system unegoes a tanscitical bifucation about the bounay point D as μ passes though μ 0. μ + a Bounay point F. Eigenvalues ae λ1 = γ +, λ =( )( μ μ ) γ 1, an λ3 = sμ. Une the assumption (3), λ 1 an λ 3 ae always negative. F is a stable noe, if μ1 > μ, λ < 0 an it is unstable if 0 < μ1 < μ, λ > 0. It can be easily seen that the system () unegoes a tanscitical bifucation as μ passes though μ 1. Next, we analyze the stability of inteio equilibium point E* =( x*, y*, z* ). About E*, the Jacobian matix J takes the fom an its eigenvalues satisfy whee J E* xf * x xf * y xf * z = yg * x yg * y yg * z zh * x zh * y zh * z λ3 + Aλ + Bλ + C = 0, (7) A γx* + y* + sμ, z* B γx** y + x** y ( a + ) ( a γ ) + sμ( γx* + y* ), ( y* ) z* ( + a) k( + a) μ C sμx** y γ +( a + )( a γ) +. ( y* ) ( y* ) To etemine the type of stability of E*, we etemine the sign of A, C,anAB C. Hee, it is clea z* that A > 0. Fom the algebaic equation F( x*, y*, z* ) = 0,wehave = γ γx* + ( a γ) y* y* y an fom G ( x*, y*, z* ) = 0,wehavex * = *,thenc can be witten as + a sμx** y C = [ a( + a γ) ay* ( + a γ) + kμ( + a) γ( a + )]. y* To etemine the sign of C, we consie the following two cases sepaately: )

8 8of4 Natual Resouce Moeling FATTAHPOUR ET AL. (a) 0 < μ1 < μ < μ0. Fom (4), we have sμx** y a ( + a γy ) * = Δ, theefoe, C = [ Δ], which is positive. y* (b) 0 < μ < μ < μ. In this case, 0 1 sμx** y y* thus C = [ Δ], which is negative. a ( + a γy ) * = + Δ, In contast by substituting z* = μ(1 ky* ) an = γ γx* + ( a γ) y* in the expession y* fo AB C, an afte some algebaic simplification, we get AB C = ( γx* + y* )( ax* y* ( a γ + ) + sμ( γx* + y* + sμ)) ( a+ ) x** y ( γx* + y* ( a+ ) y* )( γx* + y* ) ( a+ ) ksμx** y y* z* y* ax* y* ( a γ + )( γx* + y* ) + sμ ( γx* + y* + sμ)[ γx* + y* ( a + ) x* y* ] = ax* y* ( a γ + )( γx* + y* ) + sμ ( γx* + y* + sμ)[ x* ( γ ay* ) + y* (1 x* )] >0, une the assumption (3). By the Routh Huwitz citeion, all eigenvalues have negative eal pats if an only if A >0, C >0, an AB C > 0. Theefoe, E* is locally asymptotically stable if 0 < μ1 < μ < μ0 an it is unstable if 0 < μ0 < μ < μ1 We summaize the above esults as follows: Theoem 1. Let (3) hols, then the system () has one tivial equilibium O, thee axial equilibia R, N, an Q, an two bounay equilibia D an F. Moeove, (a) The tivial equilibium point O is an unstable noe, wheeas the axial equilibium points R, N, an Q ae sale points. (b) The bounay point D is a stable noe if 0 < μ0 < μ an it is a sale if 0 < μ < μ 0. (c) The bounay point F is a stable noe if 0 < μ < μ 1 an it is a sale if 0 < μ 1 < μ. () If 0 < μ1 < μ < μ0 ( 0 < k 0 < k ), then thee is a unique inteio equilibium E*, which is asymptotically stable. (e) If 0 < μ0 < μ < μ1 ( 0 < k < k 0 < 1 ), a unique inteio equilibium E* exists, which is unstable. We note that although E* changes its stability, howeve it is not possible to have a Hopf bifucation thee. Because fo that to happen, it is necessay that Equation (7) has puely imaginay oots λ =±iω an it shoul be in the following fom: λ3 + Aλ + Bλ + C = ( λ + ω)( λ + A). It is easy to veify that this hols only if B = ω an C = Aω. Because A is always positive, this implies that C >0, B >0, an AB C = 0. Howeve, we pove that une the assumption (3),

9 FATTAHPOUR ET AL. Natual Resouce Moeling 9of4 FIGURE 1 One paamete bifucation iagams fo the system () pouce by AUTO, with k = 0.6 an f as the bifucation paamete, showing a tanscitical bifucation of D an E* at f 0 = an a tanscitical bifucation of E* an F at f 1 = Othe paamete values ae a = 0.1, = 0.44, s = 0.49, γ = 0.8, = 1. All panels show f on the hoizontal axis, wheeas the vetical axes ae x, y, an z in the left, mile, an ight panels, espectively. Soli e lines coespon to banches of stable equilibia, an soli black lines to unstable equilibia. The potions of banches coesponing to x <0an y <0have no significance in the moel, but ae shown to claify the tanscitical bifucation an change of stability AB C is always positive. Then, Equation (7) cannot have puely imaginay oots. Also, fo a set of paticula paamete values, we have numeically compute AB C an use the path continuation AUTO to numeically fin the vaious bifucations. Fo example, when a= 0.1, = 0.44, s = 0.49, γ = 0.8, = 1, k = 0.6, we obtain f 0 = an f 1 = This suggests that only the tanscitical bifucation can occu. See Figue 1. Now, we consie the global behavio of the system (). Because, in ou system, all vaiables shoul be nonnegative, we ae only inteeste in the ynamics of moel () in the close fist octant + 3. Fist, we emin the efinition of a unifomly pesistent egion. Definition 1 (see Chen, 005). If thee exists a compact set Int = Ω, such that all solutions of () with initial conition eventually ente an emain in egion, then the system () is calle unifomly pesistent. Now, let us enote + 3 ={( x, y, z): 0 x 1,0 y 1,0 z 1}. (8) The vecto fiel efine by system () is locally Lipschitz continuous in, which guaantees the existence an uniqueness of thei solutions. In the following, we show that the egion efine above is unifomly pesistent une the flow of (). Lemma 1. The set in (8) is an attacting set fo the system (). Poof. Consie the system (). Fist of all, it is clea that the planes x =0, y =0, an z =0ae invaiant une the flow of system (). Now, we show that all obits with initial

10 10 of 4 Natual Resouce Moeling FATTAHPOUR ET AL. conitions ( x0, y0, z0) Ω with 1 y0 M < will eventually ente egion. To show this, we note that fo x0 1, we have x γ(1 x) x ( γ a) y < 0. This implies that thee is some finite time T 0 such that fo t T 0, we have x() t 1. Similaly, if y0 1, then y ( ) y ( + a) x = ( + ( + a) x) < 0. Theefoe, thee is some finite T 1 such that fo t T 1, we have yt () 1. Also, fo z 1 s sky z ( s f) = f < f. 1 ky 1 ky 0, Thus, thee is some finite time T so that fo t > T, z( t) 1. Pevious iscussion an the invaiance of the planes x =0, y =0, an z =0imply that the cube is an attacting set fo system (). Theoem. Suppose 0 < μ1 < μ < μ0 an assumption (3) ae satisfie, then the egion efine by Lemma 1 is unifomly pesistent une the flow of the system (). Poof. We pove this theoem by the metho of aveage Lyapunov function (cf., Ga & Hallam, 1979; Hutson, 1986). Let νx (, yz, ) = xyz α, whee α is a positive constant to be etemine. Theefoe, ν x y ξ x y z α z z (,, ) = + + = γ γx + ( a γ) y ν x y z y sz + [( ) ( + a) x y] + α s f. 1 ky Now, we consie the value of ξ ( x, y, z) at each of the bounay equilibia (limit set of () on the bounay of ) an pove that this function is positive at each of these bounay equilibia. We have ξ( O) = ξ(0, 0, 0) = γ + ( ) + ( s f), ξ( R) = ξ(0, 0, μ) = γ +, μ ( ) ( ) ξ N ξ a ( )+ γ ( ) = 0,, 0 = + α( s f), which ae positive by the assumption (3). At the equilibium Q = (1, 0, 0), we obtain ξ (1,0,0)= ( a + )+ α( s f). Fo any given set of paametes a,, s, an f,itis sufficient to choose α > a +, theefoe ξ ( Q ) is positive. Likewise, we get the values of ξ at s f equilibia D an F, espectively,

11 FATTAHPOUR ET AL. Natual Resouce Moeling 11 of 4 (A) (B) (C) FIGURE (a) Equilibium coves of coal eef an paotfish an tajectoies ove time, (b) equilibium coves of macoalgae an paotfish, an (c) the phase potait of macoalgae an coals

12 1 of 4 Natual Resouce Moeling FATTAHPOUR ET AL. ξ( D) = γ + ( a γ) μ (1 k)+ k + (1 k) + k + (1 k) + k + 0 = ( μ μ), ( ) γ( a+ ) + a ( + a ) γ μ μ ξ( F) = ( + a) 1 = ( + a) ( + a) γ γ a ( )( + )+ γ ( + ) ( μ ( (1 k)+ k) ) = + a γ = μ = ( μ μ ), 1 which ae positive. Theefoe, ν is an aveage Lyapunov function an thus, the system () is unifomly pesistent by Theoem 3.1 in Hutson (1986). 4 THE DDE MODEL AND ITS MATHEMATICAL ANALYSIS In this section, we have all assumptions about the system () an hypothesis (3) satisfie an futhe assume that 0 < μ1 < μ < μ0, (9) so that thee is a unique stable inteio equilibium. It has been suggeste by Blackwoo an Hastings (011) that the inheent time elay has significant impact on ynamics of coal algae inteactions. Hee, we constuct the following elay moel by using the fact that it takes a long time befoe algal tufs accumulate gowth afte macoalgae ae gaze by paotfish: x z( t τ) x( t τ) = x[( a γ) y + γ γx], t yt ( τ) y = y [ ( + ax ) y], t z sz = z ( s f). t 1 ky (10) In the following, we ae inteeste in stability of equilibia in the DDE moel (10). 4.1 Stability analysis of equilibia The equilibium points of the system (10) ae the same as those of the system (). The lineaize system takes the fom x t y t z t x() t x( t τ) = A1 yt () + A yt ( τ), z() t z( t τ)

13 FATTAHPOUR ET AL. Natual Resouce Moeling 13 of 4 whee γ xγ +( a γ) y ( a γ) x 0 A = ( a + ) y y ( a + ) x 0 1 ksz 0 s f (1 ky) sz 1 ky, (11) z( t) z() t x() t x() t yt ( ) ( yt ( )) yt ( ) A =. (1) Now, we analyze the stability of these equilibia. At the equilibium O = (0, 0, 0), wehave A1 = J( O) an A =[0] 3 3, then the equilibium O emains unstable fo any value of the elay τ. At the equilibium R = ( 0, 0, s f ) s, we have A1 an A as γ 0 0 μ 0 0 A1 = 0 0, A = ksμ sμ Then, the chaacteistic equation is given by μ et e = e + +. ( λi A A λτ) λ γ λτ 1 ( λ )( λ sμ) (13) μ Fo τ =0, the eigenvalues ae λ1 = γ > 0, λ = > 0, an λ3 = sμ <0. Thus, the axial equilibium R is unstable. Fo τ >0, as the time elay τ inceases, any changes in the sign of eal pat of eigenvalues coespon to a puely imaginay oot λ =i ω. Now, we assume that thee is some τ* = τ > 0 such that Equation (13) has a oot λ ()= τ α()+i τ ω()satisfying τ α ( τ* )=0, λ( τ* )=i ω( τ* )=iω. Then, μ μ μ i ω = γ e i ωτ* = γ cos ωτ* + i sin ωτ*. (14) By equating the eal an imaginay pats of (14) an using the ientity sin ωτ* + cos ωτ* = 1 we obtain, ω = μ 4γ 4 = 1 ( μ γ)( μ + γ), 4 which is negative ue to μ <γ in the assumption (3). Theefoe, Equation (13) has no puely imaginay oots, an hence, the numbe of oots with positive eal pats of Equation (13) oes not change as the time elay τ inceases (cf., Cooke & Gossman, 198). Theefoe, fo any τ >0, the lineaize equation of the system (10) at the equilibium R has one negative eigenvalue, two positive eigenvalues, an all othe eigenvalues have negative eal pats. Consequently, fo any τ >0, the axial equilibium R is unstable.

14 14 of 4 Natual Resouce Moeling FATTAHPOUR ET AL. By simila agument, we can conclue that the time elay has no effect on the stability of the equilibia N =(0,,0) an Q = (1, 0, 0). We omit the etails hee. μ ( (1 k)+ k Now, we iscuss the stability of the equilibium D =(0,, ) ). At this equilibium, A 1 an A ae ( ) γ +( a γ) 0 0 A1 = ( a + ) ( ) ( ) 0, A = 0 ksμ sμ Then, the chaacteistic equation is μ ( (1 k)+ k) ( + ) λ γ ( a γ) + ( s f)( (1 k) + k) e λτ [ λ +( )][ λ + sμ]=0. s ( + ) Hence, two of the eigenvalues ae λ = ( ) an λ3 = sμ, which ae negative [une the assumption (3)]. The othe eigenvalue satisfies ( ) λ = γ + ( a γ) e (1 k) + k ( + ) 0 = [ μ μe λτ]. μ ( (1 k)+ k) ( + ) λτ (15) (1 k) + k Recall that fo τ =0, the eigenvalues ae λ1 = ( μ μ), λ = ( ) < 0 ( + ) 0, an λ 3 = sμ <0. Thus, D is a sale point. Fo τ >0, we assume that thee exists τ* = τ > 0, then Equation (15) has a complex oot λ ()= τ α()+i τ ω() τ satisfying α ( τ* ) = 0, λ ( τ* ) = i ω( τ* ) = iω. Then, k k i ω = (1 ) + [( μ0 μ cos ωτ* ) + iμ sin ωτ* ]. (16) ( + ) Theefoe, by equating the eal an imaginay pats of (16), we obtain ω k k = ( (1 ) + ) ( + ) ( μ μ ) < 0. Again thee ae no puely imaginay oots fo Equation (15). Thus, the numbe of oots with positive eal pats of Equation (15) oes not change as the time elay τ inceases (cf., Cooke & Gossman, 198). Theefoe, fo any τ >0, the lineaize equation of the system (10) at the equilibium D has one positive eigenvalue, two negative eigenvalues, an all othe eigenvalues have negative eal pats. Theefoe, the bounay equilibium D is unstable fo any τ >0. μ At the equilibium F =(1,0, μ), A1 an A take the fom γ 0

15 FATTAHPOUR ET AL. Natual Resouce Moeling 15 of 4 A A 1 = = ( ) μ γ + μ ( a γ) 1 0 γ μ 0 a + ( a + ) 0, γ 0 ks ( μ) sμ μ μ μ 1 μ ( 1 ) ( 1 γ γ) It can be easily veifie that two of the eigenvalues ae an the thi eigenvalue satisfies λ a = ( + ) ( μ μ ), λ = sμ, γ 1 3 λ γ μ = + e λτ. μ (17) μ ( + a) Recall that fo τ =0, the eigenvalues ae λ1 = γ + <0, λ = ( μ μ )>0 γ 1, an λ3 = sμ <0. Fo τ >0, we assume that λ ()= τ α()+i τ ω()is τ a oot of Equation (17) satisfying α ( τ* )=0, λ( τ* )=i ω( τ* )=iω fo some τ * >0. Then, μ μ μ i ω = γ + μ e i ωτ* = γ + μ cos ωτ + i ωτ * sin *. (18) Theefoe, we obtain γ 3 ω = 1 [(γ μ)(3μ γ)]. (19) 4 Thus, fo μ <, Equation (17) has no puely imaginay oots, but fo μ1 < < μ < γ, thee 3 is a sequence of time elays τ < τ < < τ < τ <, with j j+1 γ τ = τ = ω { } 1 j ± ±accos 1 μ + πj, j = 0, 1,,, whee Equation (17) has a pai of puely imaginay oots λ =±iω with γ ω = 1 (γ μ)(3μ γ). (0) τ λ λ τ Now, we compute =( ) 1 fom (17) an obtain ( ) 1 e = λτ. Theefoe, λ τ λμ τ λ

16 16 of 4 Natual Resouce Moeling FATTAHPOUR ET AL. (A) (B) FIGURE 3 (a,b) Solution of the system (10), with initial conition (0.4, 0.007, 0.8) an time elay τ =3.55. DDE: elay iffeential equation; 3D: thee imensional τ λ 1 Re λ τ ωτ = Re = sin = 4 λ=±i ω λ=±iω >0, Re λ( τ) which implies that ( ) ± τ τ= τ j is positive fo j =0,1,,. Theefoe, fo 0 < τ < τ + 0,the lineaize equation at the equilibium F has one negative eigenvalue, one positive eigenvalue, an all the othes have negative eal pats. Thus, the equilibium F is unstable. When τ = τj ± ( j = 1,, ), the chaacteistic equation (17) has a pai of puely imaginay oots. Fo evey j, asthetimeelayτ inceases an passes though τ = τ ± j, the numbe of eigenvalues with positive eal pats inceases by two. Theefoe, fo τ > τ + 0, tajectoies about F ae oscillatoy an move away fom F (see Figue 3). We summaize the above iscussions in the following theoem. Theoem 3. Consie the system (10) an assume that hypotheses (3) an (9) ae satisfie. Then, (a) Fo all τ >0, the chaacteistic equation about the tivial solution O, axial solutions R, N, an Q, an the bounay point D will have no pue imaginay eigenvalues an elay has no effect on thei stability. (b) If μ < μ < γ 1, then the chaacteistic equation about F will have no pue imaginay 3 γ eigenvalues an elay has no effect on thei stability. Howeve, fo μ > > μ 3 1, thee will be asequenceoftimeelaysτ = τj ±, j = 1,, so that as the time elay τ inceases an passes though τ = τ ± j, the numbe of eigenvalues with positive eal pats inceases by two. μω μ Now, we investigate the ynamics of the elay system (10) about the inteio equilibium E* =( x*, y*, z* ).Wehavealeaypovethatunetheconition(9),this point is asymptotically stable fo τ =0. Fo, τ >0, A1 ana have the following fom, espectively,

17 FATTAHPOUR ET AL. Natual Resouce Moeling 17 of 4 A A 1 = = γ x* γ +( a γ) y* ( a γ) x* 0 ( a + ) y* y* 0, 0 ksμ sμ z* zx * * x* y* ( y* ) y* Then, the chaacteistic equation is given by whee λ3 + λ( a + a e λτ)+ λ( a + a e λτ)+( a + a e λτ)=0, (1) a γ + x* γ ( a γ) y* + sμ + y*, a = γ γx* +( a γ) y*, a ( sμ + y*)( γ + x* γ ( a γ) y*) + y* sμ +( a + )( a γ) x**, y a ( sμ + y*), a sμy*( γ + x* γ ( a γ) y*) +( a + ) sμ( a γ) x**, y a z* y* z* y* sμ zy ** y* kx** y ( a + ) sμ ( y* ) ( a+ ) x*** y z ( y* ) ( a+ ) sμx*** y z ( y* ) +. When τ =0, the chaacteistic equation (1) is given by λ3 +( a + a ) λ +( a + a ) λ +( a + a )= () By compaing with Equation (7) an accoing to the conclusion in Section, we have A ( a1 + a)>0, B ( c3 + c4 )>0, an C ( a5 + a6 ) > 0, an all thee eigenvalues ae negative. When τ >0, let λ ()= τ α()+i τ ω()be τ a oot of Equation (1) satisfying α ( τ* )=0, λ( τ* )=i ω( τ* )=iω fo some τ * >0. If λ =iω is a oot, then ω 0 an iω3 a ω a ωe i ωτ* + a i ω + a iωe i ωτ* + a + a e ωτ* = 0. (3) i By equating the eal an imaginay pats of (3) equal to zeo, we have { ( ) ω3 + a ω + a ω a sin ωτ* + a ω cos ωτ* = 0, ( ) aω+ a + aωsin ωτ* aω a cos ωτ* = (4) It follows fom (4) that ( ) ( ) ω6 + a a a ω + a a a a + a a ω + a a = (5)

18 18 of 4 Natual Resouce Moeling FATTAHPOUR ET AL. Denote ( ) ( ) A a a a, B a a a a + a a, C a a, an let ω ζ, then Equation (5) becomes hζ ( ) ζ3 + Aζ + Bζ+ C= (6) The cubic function hζ ( ) can have at most thee positive oots, which ae enote by ζi = ωi, i =1,,3with 0 < ω1 < ω < ω3. Also, fom (4), we get ( a a a ) ω4 +( a a a a + a a ) ω a a a4 ω +( a ω a6) aω5+( a + aa aa) ω3 +( aa aa) ω a4 ω +( a a ω 6 ) cos ωτ* =, sin ωτ* =. Then, coesponing to each positive oot ζi, i = 1,, 3 of hζ ( ), thee exist a sequence of time + + elays 0 < τi,0 < τi,1 < < τin, < τin, +1< such that () has a pai of pue imaginay eigenvalues, whee τ a a a ω a a a a a a ω a a = 1 4 ( ) i + ( ) i 5 6 ±accos + nπ, (7) ωi a ω +( a ω a6) ± 4 1 in, 4 i fo i = 1,, 3, n = 0, 1,,. Now, we iscuss the stability of the inteio equilibium E* accoing to the numbe of positive oots fo Equation (5). Thee ae fou cases of inteest: (I) Equation (5) has no positive oots. Then, elay has no effect on the stability of E*, which is asymptotically stable fo all τ >0. (II) hζ ( ) only has one simple positive oot ζ1 = ω1. Thee is one positive oot fo Equation (5), which is simple. Thus, fo τ = τ ± 1, n, the chaacteistic equation (1) has a pai of puely imaginay oots. To iscuss the possible Hopf bifucation about E* as τ passes though τ ± (Re λ) 1, n, we nee to etemine sign{ } ± τ τ= τ 1, n.fomequation(1),we have i λ τ 1 λτ λ a λ a a λ a = e (3 + + ) + + ( aλ + aλ+ a) λ τ aλ aλ a, e λτ = ( ) λ λ + a λ + a λ + a Thus,

19 FATTAHPOUR ET AL. Natual Resouce Moeling 19 of 4 sign (3 λ + a1λ+ a3) =sign Re + λλ ( 3 + aλ 1 + aλ 3 + a5) =sign ( ζ1 a3)(3ζ1 a3)+ a1( a1ζ1 a5) a4 + a ( a ζ a ) { ζ ( ζ a ) +( a ζ a ) a ζ +( a ζ a ) } =sign =sign{ h ( ζ )}>0, λ 1 { τ } (Re λ) { } = sign Re ± ( ) τ τ= τ1,n λ=iω1 aλ + a4 λaλ ( + aλ 4 + a6) λ=iω1 ( 1 ) ( ) ζ + a a a ζ + a a a a +a a ( aζ 1 a6) + a4 ζ1 1 because ( aζ 1 a6) + a4 ζ1 > 0. Chaacteistic equation (1) with τ =0 has thee negative eigenvalues, then when 0 < τ < τ + 1,0, the chaacteistic equation (1) has the eigenvalues with negative eal pats, an thus the equilibium E* is asymptotically stable. When τ = τ + 1,0, the chaacteistic equation (1) has a pai of puely imaginay oots ±iω1 an h ( ζ1 ) > 0, then as τ incease though τ ± 1, n, a Hopf bifucation occus, an a nontivial + peioic solution appeas. When τ1,0 < τ < τ1,1, the chaacteistic equation (1) has a pai of eigenvalues with positive eal pats, wheeas the othes with negative eal pats, hence the equilibium E* becomes unstable, an so on. Theefoe, afte a pai of imaginay eigenvalues appea, the stability of the solution can only be lost but not egaine as τ inceases. (III) If Equation (5) has a pai of positive oots with ouble multiplicity ω1 = ζ1, then h ( ζ1 ) = 0, then the tansvesality conition fo the Hopf bifucation oes not hol, thus the iection of movement of eigenvalues by inceasing τ epens on the highe eivatives, which ae vey complex. Now, suppose thee ae two positive simple oots ω1 = ζ1, ω = ζ fo Equation (5), with ω > ω 1. In this case, using Equation (4), ± thee ae the two sets of values of τ = τin,, n = 0, 1,, i = 1,, fo which Equation () has two pais of pue imaginay eigenvalues. In this case, h ( ζ1) < 0 an h ( ζ) > 0 an by simila calculation as Case (II), we have 1 1 λ λ sign Re < 0, sign Re > 0. τ τ λ=iω1 λ=iω Chaacteistic equation (1) with τ =0 has thee negative eigenvalues, an τ,0 < τ1,0 + Othewise suppose τ,0 > τ1,0 +, but h ( ζ1 ) < 0, then we shoul have a pai of complex eigenvalues with positive eal pats fo ζ < ζ 1, which is impossible, because ζ = ζ 1 is the smallest positive value, fo which hζ ( ) = 0 has a pai of pue imaginay oots. Then, when 0 < τ < τ +,0, the chaacteistic equation (1) has thee eigenvalues with negative eal pats, an thus the equilibium E* is asymptotically stable; when τ = τ +,0, the chaacteistic equation (1) has a pai of puely imaginay oots ±iω, an sign h ( ζ) is positive, then a supecitical Hopf bifucation occus, an a nontivial peioic solution exists fo + τ,0 < τ < τ1,0 +. Fo these values of τ, the chaacteistic equation (1) has a pai of eigenvalues with positive eal pats, an the othes have negative eal pats, hence the equilibium E* is unstable; when τ = τ + 1,0, the chaacteistic equation (1) will have anothe pai of puely imaginay oots ±iω1, an h ( ζ1 ) < 0, hence a subcitical Hopf + +.

20 0 of 4 Natual Resouce Moeling FATTAHPOUR ET AL. bifucation occus as τ eceases though τ + 1,0, an a nontivial peioic solution bifucates + fom the equilibium E*; when τ1,0 < τ < τ,1, the chaacteistic equation (1) has thee eigenvalues with negative eal pats, then the equilibium E* is asymptotically stable; when τ = τ,1, the chaacteistic equation (1) has a pai of puely imaginay oots ±iω, an h ( ζ ) > 0, then a supecitical Hopf bifucation occus, an a nontivial peioic solution appeas as τ inceases though fo τ = τ,1, an so on. Note that h ( ζ) > 0 an h ( ζ1 ) < 0, then afte one cycle, the numbe of eigenvalues with positive eal pats oes not change. Theefoe, if thee ae two imaginay oots, then the stability of the solution can change fo a finite numbe of times as τ inceases, an eventually it becomes unstable. (IV) Thee ae thee positive oots ω1 = ζ1, ω = ζ, ω3 = ζ3 fo Equation (5), with ω3 > ω > ω1. In this case, using Equation (4), coesponing to each pai of pue imaginay eigenvalues ωi, i = 1,, 3, thee ae two sequences of time ± elays τ = τin,, n = 0, 1,, i = 1,, 3. In this case, h ( ζ1) > 0, h ( ζ ) < 0, an h ( ζ3 ) > 0, an simila to Case (II), we have λ h ζ sign Re = sign{ ( τ i)}. ± τ= τ in, We know that chaacteistic equation (1) with τ =0has thee negative eigenvalues. With simila easoning as Case (III), the following two cases ae possible: + + (a) τ3,0 < τ,0 < τ1,0 + : When 0 < τ < τ + 3,0, the chaacteistic equation (1) has the eigenvalues with negative eal pats, thus the equilibium E* is stable; when τ τ +, the chaacteistic equation (1) has a pai of puely imaginay oots ±iω3, an = 3,0 h ( ζ3) is positive, then the Hopf bifucation occus, an a nontivial peioic solution exists fo τ = τ + + 3,0 ; when τ3,0 < τ < τ,0 +, the chaacteistic equation (1) has a pai of eigenvalues with positive eal pats, the othes with negative eal pats, hence the equilibium E* is unstable; when τ = τ +,0, the chaacteistic equation (1) has a pai of puely imaginay oots ±iω, an h ( ζ) is negative, hence the Hopf bifucation occus, + an a nontivial peioic solution bifucates fom E*; when τ,0 < τ < τ1,0 +, the chaacteistic equation (1) has the eigenvalues with negative eal pats, then the equilibium E* is stable; when τ = τ + 1,0, the chaacteistic equation (1) has a pai of puely imaginay oots ±iω1, an h ( ζ1) is positive, then the Hopf bifucation occus, an a nontivial peioic solution appeas. + + (b) τ1,0 < τ,0 < τ3,0 + : When 0 < τ < τ + 1,0, we can have simila aguments as Case (a) by exchanging inices 1 an 3. Note that the numbe of pue imaginay oots of the chaacteistic equation (1) epens on the sign of C 1 in Equation (6). If C 1 is positive, then one of the two cases (I) o (III) can occu; if C 1 is negative, then one of the two cases (II) o (IV) can occu. We summaize the above iscussions in the following theoem.

21 FATTAHPOUR ET AL. Natual Resouce Moeling 1 of 4 TABLE 1 Paamete values an efinitions Paamete Value Definition a 0.1 Rate macoalgae iectly ovegow coal (1/yea) γ 0.8 Rate macoalgae spea vegetatively ove algal tufs (1/yea) 1 Rate of coal ecuitment to algal tufs (1/yea) 0.44 Natual coal motality accounts fo 4% (1/yea), an peation accounts fo 30% (1/yea) s 0.49 Rate of paotfish gowth (1/yea) k 0.6 Dimensionless paamete that etemines the stength of the linea elationship between coal cove an caying capacity f 0.08 Rate of estuction of paotfish esulting fom fishing effot Theoem 4. Suppose ωj ζj an h ( ζj) 0 fo j =1,,3, then sign h ( ζj) = sign λ τ ± ( jn, ) τ, fo j =1,,3an n =0,1,, Moeove, if Equation (6) has one simple positive oot ζ 1 o has two positive oots, ζ1, ζ with ζ < ζ1, then thee is a Hopf bifucation fo the system (10) as τ passes though τ 1, + n, leaing to a stable peioic solution that bifucates fom E*. If thee ae thee positive oots of (6), ζ < ζ < ζ, espectively, then 1 3 ± ± λ ( τ1,n) λ ( τ, n) λ ( τ3, n) >0, <0, > 0, n = 0, 1,,, τ τ τ + which implies that thee is a Hopf bifucation as τ inceases though min{ τ1,0, τ3,0 + }, leaing to a stable peioic obit. ± 5 NUMERICAL SIMULATIONS In this section, we cay out some simulations with the following paametes: a = 0.1, = 0.44, s = 0.49, γ = 0.8, = 1, f = 0.08, k = 0.6, (8) (A) (B) FIGURE 4 (a,b) Solution of the system (10), with initial conition (0., 0.33, 0.67) an time elay τ =3.15 about the equilibium E*. DDE: elay iffeential equation; 3D: thee imensional

22 of 4 Natual Resouce Moeling FATTAHPOUR ET AL. (A) (B) FIGURE 5 (a,b) Solution of the system (10), with initial conition (0., 0.33, 0.67) an time elay τ =3.17 aoun the equilibium E*. DDE: elay iffeential equation; 3D: thee imensional (A) (B) FIGURE 6 (a,b) Solution of the system (10), with initial conition (0., 0.33, 0.67) an time elay τ =3.0 aoun the equilibium E*. DDE: elay iffeential equation; 3D: thee imensional whee we have a unique stable inteio equilibium E*. See Table 1 fo paamete values an thei efinitions (these paamete values can be foun in Mumby et al., 007; Mumby, Foste, & Glynn Fahy, 005). Fo these values of paametes, the steay states ae as follows: O = (0, 0, 0), R = (0, 0, ), N = (0, 0.56, 0), Q = (1, 0, 0), D = (0, 0.56, ), F = ( , 0, ),E * = (0.0799, , ), an the citical paamete values ae f0 = , f1 = , an k 0 =0.45. In ou analytical stuy, we know that the bounay planes { x = 0}, { y = 0}, an { z = 0} ae invaiant. Hence, the phase potaits of the system (), in the absence of macoalgae (x), coal eef (y), an paotfish (z), ae given in Figue a c, espectively. As we have seen in ou analytical analysis, the bounay point F an the unique inteio steay state E* ae the most impotant equilibium points fom the ecological point of view. Hence, we put emphasis on these in ou numeical stuy. Now, we consie the steay state F. Accoing to Theoem 1, if f1 < f < s, the bounay equilibium F is stable, but it is unstable fo f < f 1. Theefoe, fo the set of paamete values given in (8), F is unstable. Now, we consie the effect of elay, by Theoem 3, we know that if 0 f< f1 = < f = 0.867, the equilibium F is unstable fo all τ 0. Also, when τ = τ0 = 3.558, then Equation (17) has a pai of puely imaginay

23 FATTAHPOUR ET AL. Natual Resouce Moeling 3 of 4 FIGURE 7 Two paamete bifucation iagam of the system () showing cuves of coimension one local bifucation in the ( f, k) paamete plane. Also, egions in ( f, k) paamete space, whee () has positive intenal equilibia, fo fixe paamete values a = 0.1, = 0.44, s = 0.49, γ = 0.8, = 1 have been shown oots λ =±iω with ω = Tajectoies about F ae oscillating away fom F towa E* an then away fom E*. This is shown in Figue 3. By substituting the paamete values in (8) an (7), we obtain A= , B = , C = Then, by Theoem 1, the equilibium E* is stable. Now, we consie the system (10). Case II occus fo the inteio equilibium E*. Then, as we can see fom Figues 4, 5, an 6, the tajectoies about E* change as follows when τ inceases: (i) Fist, it becomes oscillatoy but eventually appoaches E* (ue to the appeaance of a pai of complex eigenvalues with negative eal pats). See Figue 4 as an illustation. (ii) By inceasing the elay τ futhe, as τ passes though τ + 1,0, a Hopf bifucation occus an a stable peioic obit bifucates fom E*. See Figue 5 as an illustation. (iii) By inceasing τ futhe, as τ passes though τ + 1,1, the peioic obit becomes unstable. See Figue 6 as an illustation. 6 DISCUSSION In this pape, we have povie local an global stability analyses of all steay states of the coal eef ODE moel () une fishing. We have pove that thee is an ecologically meaningful attacting egion, fo which the system is unifomly pesistent. We have shown that fishing plays a cucial ole on the coal eef ynamics. If the fishing ate is lowe than some theshol (f < f 0 ), the coal paotfish state is globally attacting stable noe, which implies that the eefs ae healthy. By inceasing the fishing ate (fo f > f 1 ), the macoalgae paotfish state becomes a stable noe, which implies that macoalgae aise an eefs switch fom healthy to unhealthy. When the fishing ate is in some intemeiate ange 0 < k < k 0, then the inteio state E* is

24 4 of 4 Natual Resouce Moeling FATTAHPOUR ET AL. unstable, but the coal paotfish state an the macoalgae paotfish state ae both stable; howeve, if k0 < k < 1, then the inteio state E* is stable, an the coal paotfish an macoalgae paotfish states ae both unstable (see Figue 7). In contast, because it takes some time fo algal tufs to aise afte macoalgae ae gaze by paotfish, then the inheent time elay has significant impact on the ynamics of coal algae paotfish inteactions. Theefoe, we have incopoate the elay in ou moel an teate both the fishing effot an the time elay as the bifucation paametes. Delay has no effect on the stability of the extinction state, macoalgae only state, coal only state, an paotfish only state. Howeve, stability of the coal paotfish state an the macoalgae paotfish state epens not only on the fishing effot but also on the time elay. Reefs emain healthy only fo a low fishing ate an a shot elay time. With high fishing ate an a long elay time, the macoalgae paotfish state is stable, an hence the healthy eefs switch to the macoalgae ominant status. Fo some citical values of the time elay, a Hopf bifucation occus, which leas to a nontivial peioic solution. This implies a switch between healthy an unhealthy states. This phenomenon can only appea in the DDE moel. Fo lage enough time elay, oscillations with lage amplitues appea. Finally, some numeical simulations ae caie out fo illustating the analytic esults. REFERENCES Bellwoo, D. R., Hughes, T. P., Folke, C., & Nystom, M. (004). Confonting the coal eef cisis. Natue, 49, Blackwoo, J. C., & Hastings, A. (011). The effect of time elays on Caibbean coal algal inteactions. Jounal of Theoetical Biology, 73, Blackwoo, J. C., Hastings, A., & Mumby, P. J. (01). The effect of fishing on hysteesis in Caibbean coal eefs. Theoetical Ecology, 5, Chen, F. D. (005). On a nonlinea nonautonomous peato pey moel with iffusion an istibute elay. Jounal of Computational an Applie Mathematics, 180(1), Cooke, K. L., & Gossman, Z. (198). Discete elay, istibute elay an stability switches. Jounal of Mathematical Analysis an Applications, 86, Ga, T. C., & Hallam, T. G. (1979). Pesistence in foo web 1, Lotka Voltea foo chains. Bulletin of Mathematical Biology, 41, Hughes, T. P., Bai, A. H., & Bellwoo, D. R. (003). Climate change, human impacts, an the esilience of coal eefs. Science, 301, Hutson, V. (1986). A theoem on aveage Liapunov functions. Monatshefte fü Mathematik, 98, Li, X., Wang, H., Zhang, Z., & Hastings, A. (014). Mathematical analysis of coal eef moels. Jounal of Mathematical Analysis an Applications, 416, Mumby, P. J. (006). The impact of exploiting gazes (Scaiae) on the ynamics of Caibbean coal eefs. Ecological Applications, 16, Mumby, P. J., Foste, N. L., & Glynn Fahy, E. A. (005). Patch ynamics of coal eef macoalgae une chonic an acute istubance. Coal Reefs, 4, Mumby, P. J., Hastings, A., & Ewas, H. J. (007). Theshols an the esilience of Caibbean coal eefs. Natue, 450, Panolfi, J. M., Babuy, R. H., & Sala, E. (003). Global tajectoies of the long tem ecline of coal eef ecosystems. Science, 301, How to cite this aticle: Fattahpou H, Zangeneh HRZ, Wang H. Dynamics of coal eef moels in the pesence of paotfish. Natual Resouce Moeling. 018;e10.