Journal of Theoretical Biology 248, 501-511 - University of Leicester

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Author's personal copyARTICLE IN PRESSA.Y. Morozov et al. / Journal of Theoretical Biology 248 (2007) 501–511 505v0 0uuvu cr0 uu crvhappens. Curve R corresponds to a disappearance of the‘big’ limit cycle. The position of R is found numerically bymeans of solving (5)–(6) for various parameter values andidentifying the type of the system behavior for each givenparameter set.The regimes 1 and 2 from Fig. 2 are thus characterizedby existence of a globally stable mode, i.e., a stablestationary state or a stable limit cycle, respectively. For theregimes 3 and 4, stability takes place only for a part of thephase plane. Large initial densities of phytoplankton showa further unbounded exponential increase. Finally, theregime 5 is globally unstable; all trajectories go to infinity.A typical diagram in the case when l 2 oa is shown inFig. 1b for a ¼ 0.9, l 2 ¼ 0.1, u s ¼ 10. The domain numbershave the same meaning as described above. The neutralitycurve N now lies outside of the diagram and the stationarystate is always unstable.By considering the parametric diagrams and the phaseportraits of the conceptual model (5)–(7), one can make animportant conclusion. Here we should recall that thesystem (5)–(6) without the migration term is globallyunstable for any parameter values. Introduction of themigration term changes the model properties in such a wayv1 23vu cr0 uu cr0 uu crFig. 2. The phase portraits of model I shown by using the Poincaré sphererepresentation. The portrait numbering corresponds to numbering of thedomains in Fig. 1.54that the parameter space now contains a few domainswhere the system has either a stable equilibrium or stablelimit cycle. Therefore, vertical migrations of zooplanktonfrom the deep layer can lead to stabilization of initiallyunstable plankton community.Interestingly, our model predicts that the aforesaidstabilization of plankton communities does not dependon the intensity of downward zooplankton migration. Thelatter is quantified by (dimensionless) parameter l 1 whichdoes not influence the parametric diagram properties inany significant way, only slightly affecting the size of theattractor basin.Also, the model predicts that the chance for systemstabilization is higher for l 2 4a. That can be seen from thecomparison between the diagrams in Fig. 1 where regimes 1and 2 of global stability become impossible for small l 2 .Moreover, domain 5 of global instability grows in size witha decrease in l 2 . By returning to the original parameters,from l 2 4a we obtain that L 1 K Z Z d 4r 2 0 . In ecologicalterms, it means that stability in a plankton community ismore probable for low rates of phytoplankton multiplicationand/or for high rates of upward zooplanktonmigrations (given by a product of L 1 and Z d ). The latterseems to agree very well with intuitive expectations thatvertical zooplankton migrations may act as a selforganizedecosystem response to the impact of destabilizingfactors.Moreover, stabilization of a plankton communitydepends to a large extent on the zooplankton mortalityrate g, which, in our model, is the sum of natural mortalityrate and the rate of consumption of zooplankton bytop predators (e.g., carnivorous zooplankton or planktivorousfish). The model shows that an increase inzooplankton mortality would likely result in a loss ofstability when zooplankton is not able to control algalgrowth.A question may arise here how important is the effect ofsaturation in the zooplankton migration rate, cf. the linesabove Eq. (3). In order to address this issue, we considereda particular case of the model (5)–(6) (corresponding tou s -N) where saturation is neglected. Ecologically, it maymean that saturation does not take place until thephytoplankton density reaches very high values. Weobtained that in this case the system has only threedifferent phase portraits that are qualitatively similar to theportraits 1, 2 and 5 from Fig. 2. The parametric diagram isthus missing two domains, i.e., domains 3 and 4, where thesystem is stable for not very large initial phytoplanktondensity. Moreover, for the system without saturation, thedomains corresponding to stable modes increase in size.Therefore, in agreement with the earlier inference, weconclude that saturation in the zooplankton migration ratedecreases plankton system’s stability.In conclusion of this section, we want to mention that asmooth version of the piecewise-linear parameterization (3)(or (7)) would lead to a system with essentially the samestability properties. Indeed, ‘smoothening’ of the system
Author's personal copy506ARTICLE IN PRESSA.Y. Morozov et al. / Journal of Theoretical Biology 248 (2007) 501–511(5)–(6) would imply applying a (small) structured perturbationto function (3) in a small vicinity of u ¼ u cr andu ¼ u s ; however, it is well-known that steady states andsimple limit cycles are robust with respect to smallperturbations of the right-hand side of the correspondingequations, cf. (Kuznetsov, 1995).4. Model II: vertical migration in shallow waters4.1. Main equationsThe model considered in the previous section isessentially based on the assumption that some zooplanktonspecies mostly live at considerable depth and only ascendto feed when the phytoplankton density in the upper layeris sufficiently high. In this section, we consider dynamics ofa plankton community where all zooplankton species feedin the upper layer but may perform diurnal verticalmigrations into deep waters to avoid predation. Thissituation seems to be more relevant in shallow lakes orshallow parts of the ocean.We do not assume any more that the whole water mass issplit into two layers. Instead, in order to quantify themigratory behavior of zooplankton, we introduce a factorZ which gives the proportion of time spent by zooplanktersin the upper part of the water column. Obviously, 0pZp1;in particular, Z ¼ 1 means that all zooplankton remainsconcentrated in the subsurface waters and does notperform vertical migrations at all. The case Z ¼ 0 meansthat all zooplankton has descended to the depth and ceasedits migration for a while, e.g., due to insufficient amount offood at the top of the water column. An intermediate valueZE0.5 corresponds to the case when zooplankton spendsone half of the day in the upper waters and the other half inthe deep waters.Since there is considerable evidence that the intensity ofzooplankton migrations is positively correlated with theavailability of food, cf. (George, 1983; Hoenicke andGoldman, 1987; Dini and Carpenter, 1992; Sekino andYamamura, 1999), we assume that Z is a monotonicallyincreasing function of P. Therefore, values of Z close to 0and close to 1 should correspond to relatively low andrelatively high density of the phytoplankton density in thesubsurface waters, respectively.We then introduce a new variable V ¼ ZZ where Z is thezooplankton density averaged over the whole watercolumn. Since phytoplankton is mostly concentrated atthe top, the consumption of phytoplankton by zooplanktonis now described as f ðP; ZÞ ¼K Z ZZP=ðP þ HÞ ¼K Z VP=ðP þ HÞ, i.e., it depends both on the total amountof zooplankton Z and on its presence Z in the upper layer.It means that the left-hand side of the equation for thephytoplankton density, cf. (1), depends on V rather thanon Z. Correspondingly, the generic system (1)–(2) can nowbe re-written in new variables P and V, provided we areable to derive an equation for V.The rate of change in V is given asdVdt ¼ dðZZÞ ¼ Z dZdt dt þ Z dZdt , (9)where dZ/dt ¼ (dZ/dP)(dP/dt) so that, making use ofEq. (1),Z 0 ZZPðtÞ ¼ r 0 P K ZP þ HZ 0 ðPÞ. (10)The equation for dZ/dt is essentially Eq. (2); however,we should now take into account that, in the subsurfacewaters, zooplankton predation by higher predators cannotbe neglected:dZdt ¼ kK PVZP þ H m 0Z m 1 V. (11)Here m 0 Z and m 1 V are the zooplankton natural mortalityand its mortality due to the predation by top predator(s),respectively. The latter is assumed to depend on the timethat zooplankton spends in the upper waters and thusdepends on V rather than on Z. It should be alsomentioned that, in a general case, predation by a toppredator is described by a nonlinear function (Edwards andYool, 2000); however, a linear function still can be a goodapproximation when the zooplankton density is not veryhigh.Since in eutrophic ecosystems zooplankton mortality ismostly caused by its predation by a higher predator(Jankowski et al., 2005), we can neglect the mortality termm 0 Z for simplicity. From (9)–(11), we then obtaindPdt ¼ r VP0P K ZP þ H , (12)dVdt ¼ k K ZVPP þ Hm 1V ZðPÞþV r 0 P K ZVPZ 0 ðPÞ=ZðPÞ.P þ H(13)Note that in the particular case Z(P)1 (i.e. whenmigrations are ‘switched off’ and all zooplankton isconcentrated in the upper waters) Eqs. (12)–(13) areidentical to the standard model (1)–(2) provided the sameassumptions are made that the per capita phytoplanktongrowth is density independent and grazing is parameterizedas (4). In a more general case Z(P)consto1, which meansthat zooplankton performs diurnal vertical migrations buttheir intensity is not affected by food availability,Eqs. (12)–(13) coincide with the model (1)–(2) up to aconstant factor in the equation for the zooplanktondensity; in both cases the system remains globally unstable.Introducing, for convenience, dimensionless variablesP ¼ r 0Hu; V ¼ r 0Hwand t ¼ t ,kK Z K Z r 0from (12) to (13) we arrive at the following system:dudt ¼ uuwua þ 1 , (14)
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