A Resolution of the Paradox of Enrichment

Home , Limit cycle, Paradox of enrichment

June 9, 2015 9:12 WSPC/S0218-1274 1550094

International Journal of Bifurcation and Chaos, Vol. 25, No. 6 (2015) 1550094 (10 pages) c World Scientiﬁc Publishing Company DOI: 10.1142/S0218127415500947

Z. C. Feng Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA [email protected] Y. Charles Li Department of Mathematics, University of Missouri, Columbia, MO 65211, USA [email protected]

Received January 19, 2015

The paradox of enrichment was observed by Rosenzweig [1971] in a class of predator–prey models. Two of the parameters in the models are crucial for the paradox. These two parameters are the prey’s carrying capacity and prey’s half-saturation for predation. Intuitively, increasing the carrying capacity due to enrichment of the prey’s environment should lead to a more stable predator–prey system. Analytically, it turns out that increasing the carrying capacity always leads to an unstable predator–prey system that is susceptible to extinction from environmental random perturbations. This is the so-called paradox of enrichment. Our resolution here rests upon a closer investigation on a dimensionless number H formed from the carrying capacity and the prey’s half-saturation. By recasting the models into dimensionless forms, the models are in fact governed by a few dimensionless numbers including H. The effects of the two parameters: carrying capacity and half-saturation are incorporated into the number H. In fact, increasing the carrying capacity is equivalent (i.e. has the same effect on H) to decreasing the half-saturation which implies more aggressive predation. Since there is no paradox between more aggressive predation and instability of the predator–prey system, the paradox of enrichment is resolved. The so-called instability of the predator–prey system is characterized by the existence of a stable limit cycle in the phase plane, which gets closer and closer to the predator axis and prey axis. Due to random environmental perturbations, this can lead to extinction. We also further explore spatially dependent models for which the phase space is infinite-dimensional. The spatially independent limit cycle which is generated by a Hopf bifurcation from an unstable steady state, is linearly stable in the infinite-dimensional phase space. Numerical simulations indicate that the basin of attraction of the limit cycle is riddled. This shows that spatial perturbations can sometimes (neither always nor never) remove the paradox of enrichment near the limit cycle!

Keywords: The paradox of enrichment; predator–prey model; dimensionless number; limit cycle; spatiotemporal dynamics.

1. Introduction Choi & Pattent, 2001; Diehl, 2007; Freedman & The paradox of enrichment was ﬁrst observed Wolkowicz, 1986; Genkai-Kato & Yamamura, 1999; by Rosenzweig [1971] in a class of mathematical Jensen & Ginzburg, 2005; Kirk, 1998; Mougi & predator–prey models. Since then, there have been Nishimura, 2008; Rall et al., 2008; Riebesell, 1974; a lot of studies on the subject [Gilpin & Rosenzweig, Roy & Chattopadhyay, 2007; Scheﬀer & De Boer, 1972; Abrams & Walters, 1996; Boukal et al., 2007; 1995; Sharp & Pastor, 2011; Vos et al., 2004]. These

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Z. C. Feng & Y. C. Li studies cover a wide spectrum of topics including Equations (1)and(2)canberewritteninthe invulnerable prey, unpalatable prey, prey toxicity, following dimensionless form: induced defense, spatial inhomogeneity, etc. The ∂u ∂v ∂u ∂2u u paradox roughly says that in a predator–prey sys- − c1 = + u(1 − u) − v, (3) tem, increasing the nutrition to the prey may lead to ∂t ∂x ∂x ∂x2 u + H an extinction of both the prey and the predator. It ∂v ∂u ∂v ∂2v u is possible that the paradox is purely an artifact of + c2 = + k − r v, (4) ∂t ∂x ∂x ∂x2 u + H the mathematical models, while in reality increasing the nutrition never leads to extinction. Our studies where u = U/b, v = Vγ/(αb), t = αT , here totally focus upon the mathematical models x = X α/D, and the dimensionless numbers are themselves. We are not exploring the experimen- given by tal aspect of the subject. As far as the original h µ κγ mathematical models [Rosenzweig, 1971]arecon- H = ,r= ,k= , cerned, we notice that the paradox can be resolved b κγ α (5) once the models are put into dimensionless forms. αb b c C ,c C . In dimensionless forms, the essential functions of 1 = 1 γD 2 = 2 D control parameters can be revealed. We name H: the capacity-predation number, and r 2. The Predator–Prey Model : the mortality-food number. These two dimensionless numbers are crucial in our resolution of the The predator–prey model is as follows (a spatio- paradox of enrichment. The spatial domain is cho- temporal extension of one of those in [Rosenzweig, sen to be finite x ∈ [0,L]. Three types of boundary 1971]), conditions can be posed, ∂U ∂V ∂U − C (1) Neumann boundary condition, 1 ∂T ∂X ∂X ∂u ∂v 2 ∂ U U U ∂x = ∂x =0; = D + αU 1 − − γ V, (1) x=0,L x=0,L ∂X2 b U + h (2) periodic boundary condition, u and v are peri- ∂V ∂U ∂V C odic in x with period L; ∂T + 2 ∂X ∂X (3) Dirichlet boundary condition, ∂2V U = D + κγ − µ V, (2) u|x=0,L = v|x=0,L =0. ∂X2 U + h In the Dirichlet boundary condition case, the spa- where U is the prey density, V is the predator tially uniform dynamics [∂x =0in(3)and(4)] T X density, is the time coordinate, is the one- is excluded. Thus the original paradox of enrich- C C dimensional space coordinate, 1 and 2 are the ment for the uniform dynamics posed by Rosen- D coefficients of migration due to predation, is the zweig [1971] is also excluded. spreading (diffusion) coefficient of the species (chosen to be the same for both predator and prey), α is the maximal per capita birth rate of the prey, b is 3. Formulation of the Paradox the carrying capacity of the prey from the nutrients, of Enrichment h is the half-saturation prey density for predation, The paradox of enrichment was originally formu- γ is the coefficient of the intensity of predation, κ lated by Rosenzweig [1971] for the spatially uniform is the coefficient of food utilization of the preda- ∂ µ dynamics [ x =0in(1)and(2)]: tor, and is the mortality rate of the predator. The last two terms in Eq. (1), i.e. the prey birth dU U U = αU 1 − − γ V, (6) term and the predation term, can take many differ- dT b U + h ent specific forms, but have the same characteristics as (1), which leads to the paradox of enrichment, see dV U = κγ − µ V. (7) [Rosenzweig, 1971]. dT U + h

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A Resolution of the Paradox of Enrichment

The paradox focuses upon the linear stability of the steady state given by U U 1 κγ − µ =0,α1 − − γ V =0. U + h b U + h It turns out that when other parameters are ﬁxed, increasing b leads to the loss of stability of this steady state, in which case, a limit cycle attractor around the steady state is generated. As b increases, the limit cycle gets closer and closer to the V -axis. That is, along the limit cycle attractor, the prey population U decreases to a very small value. Under the ecological random perturbations, U can reach 0, i.e. extinction of the prey. With the extinction of the prey, the predator will become extinct soon. On the Fig. 1. The phase plane of the spatially uniform system other hand, increasing b means increasing the carry- (8)and(9). ing capacity of the prey, which can be implemented by increasing the prey’s nutrients, i.e. enrichment which is the intersection point of the parabola and of the prey’s environment. Intuitively, increasing b the vertical line (Fig. 1): should enlarge the prey population and make it 2 2 1 1 more robust from extinction. This is the paradox (P) v = − u − (1 − H) + (1 + H) , of enrichment. 2 2 r u H. (V) = − r 4. The Resolution of the Paradox 1 du of Enrichment On the parabola (P), dt = 0, and on the vertical line (V), dv = 0. Linearizing the system (8)and(9) In order to resolve the paradox of enrichment, it is dt at the steady state (10), we get fundamental to rewrite the system (6)and(7)in the dimensionless form [∂x =0in(3)and(4)]: du u∗ = [(1 − 2u∗ − H)u − v], du u dt u∗ + H = u(1 − u) − v, (8) dt u + H dv kH(1 − u∗) = u. dv u dt u∗ + H = k − r v (9) dt u + H The eigenvalues of this linear system is given by 1 and the key to the resolution is a complete under- λ = (1 − 2u∗ − H) 2 standing of the dimensionless capacity-predation number H. 2 1 1 − u∗ First, we need to understand the dynamics ± − u − H − kH . (1 2 ∗ ) u (11) of (8)and(9) in detail. For all parameter values, 2 ∗ there are two trivial steady states: The sign of the real part of λ is decided by the sign of the quantity 1−2u∗ −H. Setting 1−2u∗ −H =0, (I) u∗ =0,v∗ =0; (II)u∗ =1,v∗ =0. one gets The steady state (I) is a saddle for all parameter 1 values. The steady state (II) is a stable node when u∗ = (1 − H), 1 1 2 H>r − 1, a saddle when 0

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Z. C. Feng & Y. C. Li linearly unstable. If u∗ in (10) is to the right of the symmetry axis of the parabola (P), then the steady state (10) is linearly stable. If u∗ in (10)isonthe symmetry axis of the parabola (P), then the eigenvalues λ in (11) are purely imaginary. Using this fact, we derive the following linear instability criterion of the steady state (10): 2 0

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A Resolution of the Paradox of Enrichment

The eigenvalues of this linear system is given by b b 2 1 1 λ = 1 − 2u∗ − f (bu∗)u∗(1 − u∗) + 1 − 2u∗ − f (bu∗)u∗(1 − u∗) − kbf (bu∗)u∗(1 − u∗). 2 r 2 r (15)

The sign of the real part of λ is decided by the sign of the quantity Proof. First we set up the compact region R as in Fig. 3. P1 is the steady state (10) which is unstable 1 b 1, such that the vector field is transversal to U AB and points leftward by Eq. (8). The vertical thus, for small h, the model f(U)= represents 1 U+h coordinate of B is chosen to be greater than ,so a much more aggressive predation than f(U)=U 4r BC when the prey population U is small. As U → +∞, that on the orbit , the right-hand side of (8)is negative, where the horizontal coordinate of C is U the u∗ given in (10), i.e. C and P1 have the same → 1,U→ +∞. u U + h horizontal coordinate. Notice that u+H is strictly monotonically increasing in u. The right-hand side f U U That is, the model ( )= represents unlimited of (9)atC is zero. The right-hand side of (9)on U predation ability for large prey population .In the orbit BC (except at the point C) is positive. f U U this sense, ( )= U+h serves as a better model. CD is a horizontal segment on which the v coordi- On the other hand, the prey population is finite nate is a constant, and the right-hand side of (9)is b with capacity . With a proper choice of the coeffi- negative (except at the point C). The region R is γ f U U cient of predation (6), ( )= still represents a defined as the region outside S and inside the loop limited predation ability for prey population U near ABCDP 2P3A. The region R is a positively invariant f U U its capacity. When ( )= , the eigenvalue (15) region (i.e. invariant as time increases). By the well- becomes ω known Poincaré–Bendixson theorem, the -limit set r r 2 r λ = − ± − kr 1 − , 2b 2b b where r

6. The Limit Cycle in the Phase Plane Returning to the spatially uniform system (8)and (9), we can prove the following ω-limit set theorem.

Theorem 1. Under the dynamics of (8) and (9), 2 when 0

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Theorem 2. If all the ω-limit sets of points in the ﬁrst open quadrant of the phase plane except the unstable steady state (10) is actually the same stable limit cycle S∗, then S∗ loops around the unstable steady state (10).

Proof. Let W be a small tubular neighborhood of the limit cycle S∗, which is the local stable manifold of S∗,seeFig.4.LetS1 be a closed curve in the region R and near S (Fig. 3), that loops around the steady state P1 once [in fact, S1 can be just S]. For any q ∈ S1, there is a segment neighborhood of q in T S1, ξq ⊂ S1 and a time Tq, such that F q (ξq) ⊂ W

Fig. 4. The setup for the proof that the limit cycle loops around the steady state. of any point in the region R is a periodic orbit. Since the vertical line AB can be moved to the right arbitrarily, the point B can be moved up arbitrarily, and the closed curve S can be arbitrarily small, the claim of the theorem is proved.

Remark 6.1. As shown later, it can be veriﬁed numerically that all the ω-limit sets of points in the ﬁrst open quadrant of the phase plane except the unstable steady state (10) is actually the same stable limit cycle. Proving such a claim is not easy. (a) H =0.2

Fig. 5. The limit cycle attractor in the ﬁrst open quadrant (b) H =0.1 of the phase plane, where r =1/2, H =0.3, k =1,andthe two initial conditions are (u =1.01, v =0.02) and (u =0.30, Fig. 6. The deformation of the limit cycle as H is decreased, v =0.42). where r =1/2andk =1.

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A Resolution of the Paradox of Enrichment ∂u ∂2u Hv∗(t) = + 1 − 2u∗(t) − u ∂t ∂x2 (u∗(t)+H)2 u∗(t) − v, (16) u∗(t)+H ∂v ∂2v kHv∗(t) = + u ∂t ∂x2 (u∗(t)+H)2 u∗(t) + k − r v. (17) u∗(t)+H Using the Fourier mode iξx iξx u = uξe +c.c.,v= vξe +c.c., the linear system (16)and(17) is transformed into ∗ ∂uξ Hv (t) Fig. 7. The time series graph of the limit cycle in Fig. 6(b), −ξ2u − u∗ t − u = ξ + 1 2 ( ) ∗ 2 ξ where H =0.1, r =1/2, and k =1. ∂t (u (t)+H) u∗(t) F t − vξ, where is the evolution operator of the system (8) u∗(t)+H and (9).Allsuchsegmentsformanopencoverof S S ∂v kHv∗ t 1. By the compactness of 1, there is a finite cover ξ −ξ2v ( ) u = ξ + ∗ 2 ξ {ξqn }n=1,...,N .Let ∂t (u (t)+H) ∗ T =max{Tqn }, u (t) n=1,...,N + k − r vξ. u∗(t)+H T then F (S1) ⊂ W . Since the region R is positively T A further change of variables invariant, F (S1) still loops around the steady state −ξ2t −ξ2t P1, then the tubular neighborhood W also loops uξ = e uˆξ,vξ = e vˆξ, P S around the steady state 1, thus the limit cycle ∗ transforms this linear system into P1 also loops around the steady state . ∗ ∂uˆξ ∗ Hv (t) = 1 − 2u (t) − uˆξ Numerically one can verify that the attractor ∂t (u∗(t)+H)2 in the first open quadrant of the phase plane is a ∗ u (t) limit cycle as shown in Fig. 5.AsH is decreased, − vˆξ, u∗(t)+H the limit cycle is quickly getting closer and closer to v u ∗ ∗ the -axis (and -axis) as shown in Fig. 6.Thetime ∂vˆξ kHv (t) u (t) = uˆξ + k − r vˆξ; series graph of the limit cycle in Fig. 6(b) is shown ∂t (u∗(t)+H)2 u∗(t)+H in Fig. 7. One can see clearly that with small random environmental perturbations, the system will which is a stable linear system since the limit cycle become extinct! is linearly stable in the plane. Since the plane is a subspace of the infinite-dimensional phase space under Neumann or periodic boundary condition, 7. Spatial Dependence the limit cycle is linearly stable in such an infinite- 7.1. The limit cycle is linearly dimensional phase space. stable 7.2. The riddled basin of attraction Let S∗ be the limit cycle on the plane, of the limit cycle ∗ ∗ S∗ : u = u (t),v= v (t). Since it is linearly stable, the limit cycle on The period of S∗ is T∗. Linearizing (3)and(4)at the plane is an attractor in the entire infinite- S∗,weget dimensional phase space. Numerical simulations are

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Fig. 8. An illustration of the riddled basin of attraction of the limit cycle on the plane, where the uniform, traveling wave, and chaos asymptotic states are shown in Figs. 9–11. conducted on the system (3)and(4) under the Using the Fourier mode periodic boundary condition with spatial period iξx iξx u = uξe +c.c.,v= vξe +c.c., L = 300, where c1 =0,c2 =0,r =0.8, H =0.1, and k =1.0. The initial conditions of the numerical the linear system (18)and(19)istransformedinto simulations are given by ∂u πx ξ −ξ2u r − u − H u − v , u x . ,vx . , ∂t = ξ + [(1 2 ∗ ) ξ ξ] ( )=04018 ( )=03754 + cos L ∂v − u for diﬀerent values of .When = 0, the initial con- 2 1 ∗ = −ξ vξ + rkH uξ. dition lies on the limit cycle on the plane. Figure 8 ∂t u∗ illustrates the riddled nature of the basin of attraction of the limit cycle on the plane. This riddled nature indicates that spatial perturbations sometimes (but neither always nor never) can remove the paradox of enrichment near the limit cycle on the plane.

7.3. Other bifurcations? Linearizing (3)and(4)atthesteadystate(10), we get ∂u ∂2u r − u − H u − v , ∂t = ∂x2 + [(1 2 ∗ ) ] (18) ∂v ∂2v − u rkH 1 ∗ u. Fig. 10. The traveling wave asymptotic state referred to in = 2 + (19) ∂t ∂x u∗ Fig. 8.

Fig. 9. The uniform asymptotic state referred to in Fig. 8. Fig. 11. The chaos asymptotic state referred to in Fig. 8.

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A Resolution of the Paradox of Enrichment

The eigenvalues λ of this system satisfy the plane is linearly stable for all these parameter 2 2 values. λ ξ − r − u − H λ 2 2 +[2 (1 2 ∗ )] As mentioned above, the minimum of ξ [ξ − 2 2 2 r(1 − 2u∗ − H)] in ξ also occurs at (22), where the + ξ [ξ − r(1 − 2u∗ − H)] minimal value is 2 1 − u∗ + r kH =0. (20) 1 2 2 u − r (1 − 2u∗ − H) . ∗ 4 A possible Hopf bifurcation often occurs when the A Turing bifurcation may start to occur at λ coeﬃcient of the term is zero, i.e. − u 1r2 − u − H 2 r2kH 1 ∗ , 2 (1 2 ∗ ) = 2ξ − r(1 − 2u∗ − H)=0; 4 u∗ that is while a possible Turing bifurcation often occurs 2 when the constant term is zero, i.e. 1 r 1+r k = 1 − H , (23) 4 1 − r − rH 1 − r 2 2 2 1 − u∗ ξ [ξ − r(1 − 2u∗ − H)] + r kH =0. u∗ where λ2 =0.When 2 Notice also that when the coeﬃcient of the λ term 1 r 1+r k< 1 − H , is zero, the following part (of the constant term) 4 1 − r − rH 1 − r 2 2 ξ [ξ − r(1 − 2u∗ − H)] further Turing bifurcations may occur at

ξ2 2 1 is minimal in . ξ r − u∗ − H 2 = (1 2 ) The minimum of [2ξ − r(1 − 2u∗ − H)] in 2 ξ2 occurs at ξ2 = 0, where the minimal value is 2 − u −r − u − H 1 2 1 ∗ (1 2 ∗ ). A Hopf bifurcation starts to ± r(1 − 2u∗ − H) − r kH occur at 2 u∗ 2 1 1+r 1 − 2u∗ − H =0, i.e. H = − 1; (21) = r 1 − H 1+r 2 1 − r as shown before in (12). At this Hopf bifurcation, 2 1 1+r ± r 1 − H − rk(1 − r − rH). 1 − u∗ 2 1 − r λ = ±ir kH . u∗ Numerically we did not observe any Turing bifur- The steady state (10) bifurcates into the limit cycle cation. Usually Turing bifurcations are observed (Fig. 5) on the plane. in (spatially) high-dimensional systems or one- 2 When 1 − 2u∗ − H<0 (i.e. H>1+r − 1), dimensional systems with variable coeﬃcients. by (20), the steady state (10) is linearly stable in the inﬁnite-dimensional phase space. Therefore, further 8. Conclusion bifurcation may only occur when 1 − 2u∗ − H>0 2 2 (i.e. H<1+r −1). When H<1+r −1, further Hopf In this article, we present a resolution on the para- bifurcations may occur at dox of enrichment based upon the dimensionless form of the mathematical model. We also explore 2 1 1 1+r ξ = r(1 − 2u∗ − H)= r 1 − H , (22) spatial perturbations. The conclusion is that spa- 2 2 1 − r tial perturbation sometimes (but neither always nor never) can remove the paradox of enrichment near when the limit cycle on the plane. 2 1 r 1+r k> 1 − H . 4 1 − r − rH 1 − r References But numerically we did not observe such a bifurca- Abrams, P. & Walters, C. [1996] “Invulnerable prey and tion; the reason seems to be that the limit cycle in the paradox of enrichment,” Ecology 77, 1125–1133.

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