Vibrational resonance in groundwater-dependent plant ecosystems
aa r X i v : . [ n li n . C D ] A p r Vibrational resonance in groundwater-dependent plantecosystems
C. Jeevarathinam a , S. Rajasekar a , Miguel A.F. Sanju´an b a School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamilnadu, India b Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de F´ısica,Universidad Rey Juan Carlos, Tulip´an s/n, 28933 M´ostoles, Madrid, Spain
Abstract
We report the phenomenon of vibrational resonance in a single species and atwo species models of groundwater-dependent plant ecosystems with a bihar-monic oscillation (with two widely different frequencies ω and Ω, Ω ≫ ω ) ofthe water table depth. In these two systems, the response amplitude of thespecies biomass shows multiple resonances with different mechanisms. Theresonance occurs at both low- and high-frequencies of the biharmonic force.In the single species bistable system, the resonance occurs at discrete val-ues of the amplitude g of the high-frequency component of the water table.Furthermore, the best synchronization of biomass and its carrying capac-ity with the biharmonic force occurs at the resonance. In the two speciesexcitable and time-delay model, the response amplitude ( Q ) profile showsseveral plateau regions of resonance, where the period of evolution of thespecies biomass remains the same and the value of Q is inversely propor-tional to it. The response amplitude is highly sensitive to the time-delay Email addresses: [email protected] (C. Jeevarathinam ), [email protected] (S. Rajasekar ), [email protected] (MiguelA.F. Sanju´an )
Preprint submitted to Ecological Complexity and Accepted for publication (in 2013)November 8, 2018 arameter τ and shows two distinct sequences of resonance intervals with adecreasing amplitude with τ . Keywords:
Groundwater dependent plant ecosystemsBiharmonic water table depthVibrational resonance
1. Introduction
The study of the response of a system to a small variation of environmen-tal changes is important since environmental drivers often fluctuate. Thefluctuation can be periodic or nonperiodic (noise). In certain ecosystems,a small change in one or more environmental parameters leads to consider-able changes on their structure and function. It has been noted that manysystems have relatively high levels of diversity for an intermediate level of adisturbance (Roxburgh et al., 2004). The impact of the environmental varia-tions/fluctuations has been analysed on food web stability (Vasseur and Fox,2007), species coexistence in Holt-McPeek systems (Lai and Liu, 2005) andthe stability of recovery (Steneck et al., 2002). Without invoking interactionbetween environmental noise and competition, it has been shown that envi-ronmental fluctuations enhance coexistence of species which either prefer ortolerate different environmental conditions (D’Odorico et al., 2008).In the present work, we consider groundwater-dependent plant ecosys-tems. It is important to analyse the influence of the variation of variousenvironmental factors, particularly, the changes in the water table depth, inorder to get a deep understanding of the response of various ecosystems. We2oint out that vegetation-water table interactions is very common in manyecosystems like wetlands, salt marshes and riparian forests. As a matter offact, it is considered as one of the key mechanisms influencing the dynamics ofvegetation (Naumburg et al., 2005; Elmore et al., 2006; Munoz-Reinoso andde Castro, 2005). Appropriate theoretical models are of great use for explor-ing various possible dynamics that can emerge from vegetation-water tableinteractions. In this connection, (Ridolfi et al., 2006, 2007) have proposedtwo vegetation-water table models based on realistic ecological assumptions.The first model describes the vegetation biomass dynamics of only one species(dominant species). In this model the rate of change of the species biomassdepends on the existing biomass and the carrying capacity of the system.The resultant model is a first-order nonlinear ordinary differential equationwith a periodic driver. It accounts for multistable states in the dynamics ofwetland forests and riparian ecosystems (Scheffer et al., 2001). The secondmodel describes the two phreatophyte species interacting with a water table.In these two models, one of the factors that can change the carrying capacityof biomass is the depth of the water table. Change in the water depth dueto seasonal rainfall oscillations and other sources is represented by a periodicfunction of time. It is also found to display coexistence of two species andchaotic dynamics (Ridolfi et al., 2007).The influence of the environmental variability, treated as a disturbanceor a kind of noise, in the above two models has been analyzed recently byBorgogno and his co-workers (Borgogno et al., 2012). Specifically, theyhave shown the occurrence of stochastic and coherence resonances. Whena bistable or an excitable system driven by a weak periodic force is subjected3o an additive noise, it can exhibit an enhanced response at an optimalnoise intensity. This phenomenon is termed as stochastic resonance (Gam-maitoni et al., 1998; McDonnell et al., 2008). Very recently, noise-inducedspatio-temporal patterns in wetland vegetation dynamics have been reported(Scarsoglio et al., 2012). In a subthreshold excitable system, a noise-inducedresonance can be realized in the absence of external periodic driving and isknown as coherence resonance (Pikovsky and Kurths, 1997). Interestingly,it has been shown that deterministic resonances can be observed even inmonostable nonlinear systems driven by a biharmonic force in the absenceof external noise and is called vibrational resonance (Landa and McClin-tock, 2000). The analysis of vibrational resonance has received a great dealof attention in recent years. Particularly, its occurrence has been investi-gated in a spatially extended system in the presence of noise (Zaikin et al.,2002), Duffing oscillator (Blekhman and Landa, 2004), two-coupled over-damped anharmonic oscillators (Gandhimathi et al., 2006) and monostablesystems (Jeyakumari et al., 2009). Experimental evidence of vibrational res-onance was demonstrated in analog simulations of the overdamped Duffingoscillator (Baltanas et al., 2003), a bistable optical cavity laser (Chizhevskyand Giacomelli, 2006) and an excitable electronic circuit with Chua’s diode(Ullner et al., 2003). The influence of time-delayed feedback on vibrationalresonance was studied numerically (Yang and Liu, 2010) and theoretically(Jeevarathinam et al., 2011). Further, biharmonic force induced enhancedsignal propagation was found to occur in one-way coupled systems (Yao andZhan, 2010) and in a coupled network of excitable neuronal systems (Yu etal., 2011). 4n the present work, we consider the two groundwater-dependent plantecosystem models of (Ridolfi et al., 2006, 2007) and investigate the emer-gence of vibrational resonance. It has been pointed out that the dynamicsof vegetation have a time scale greater than one season and much greaterthan man-induced periodic disturbances (Ridolfi et al., 2007). We wish tomention that high-frequency oscillation of beach water table due to waverunup and rundown has been observed and analysed (Waddell, 1976; Li etal., 1997). Interestingly, similar high-frequency oscillation of undergroundwater table (in addition to the low-frequency periodic oscillation of watertable due to seasonal variation) can occur due to evaporation, inflow andoutflow of water and temperature fluctuation. It can also be artificially real-ized through irrigation or pumping from the aquifer. Furthermore, a watertable rise and drop can be induced by vegetation removal and planting re-spectively. Thus, planting additionally short-lived species interacting weaklywith species A can also lead to a high-frequency variation of the water depth.Therefore, it is realistic to include a biharmonic force in the water table withtwo well-separated frequencies. The first model describing the dynamics ofthe biomass V of a single species has bistable states. When the biharmonicforce is included in the water depth the system shows an oscillatory variationof V . As the amplitude of the high-frequency force is varied, the system ex-hibits multiple vibrational resonance with a decreasing response amplitudeat successive resonances for certain range of fixed values of amplitude of low-frequency force. The second model describes the dynamics of two species,say A and B , interacting with the water table, and where the evolution of B depends on A ( t − τ ) and τ is the time-delay parameter. Unlike the single5pecies model, the two species model is an excitable system (such a systemhave only one stable equilibrium state, but external perturbations above acertain threshold can induce large excursions in phase space, which takesthe form of spikes or pulses). For a fixed time-delay, both A and B displaya certain number of resonances when the amplitude of the high-frequencyforce is varied. The resonance profiles of A and B are similar except that atresonance the amplitude of A is always much higher than that of B . Herethe resonance intervals are not sharp but wide. The response amplitude isinversely proportional to the period of the variation of A and B . The delayparameter τ has a strong influence on the response amplitude. The responseamplitude at successive alternative resonances decreases when the value ofthe delay parameter increases.
2. Vibrational resonance in a single species model
To start with, first we briefly introduce the model (Ridolfi et al., 2006,2007) in order to prepare the readers for the study of vibrational resonance.
The dynamics of phreatophyte biomass V of a single species (or totalplant biomass neglecting interspecies interactions) is expressed as (Ridolfi etal., 2006, 2007) d V d t = V ( V cc − V ) , (1)where the growth rate of V is assumed to be proportional to the existingbiomass and the available resources V cc − V with V cc being the carryingcapacity of the ecosystem, that is, the maximum amount of vegetation sus-tainable with the available resources. Based on experimental evidences, an6ppropriate form of V cc shows a quadratic dependence on the water tabledepth d . Taking into the effect of periodic oscillations in the rainfall regionsleading to periodic variations of water table depth, (Borgogno et al., 2012)considered the form of V cc as V cc = a [ d ( t ) − d inf ] [ d sup − d ( t )] , if d inf < d < d sup , otherwise . (2)The form of V cc given by Eq. (2) corresponds to the case of phreatophytevegetation that depends on water uptake from the groundwater. In Eq. (2) d ( t ) is the water table depth, a is the sensitivity of carrying capacity tochanges in the water table depth, d inf is the threshold of vegetation toleranceto shallow water tables and insufficient aeration of the root zone and d sup is the threshold of water depth below which tap-roots cannot extract water.The water table depth is given by d ( t ) = d + βV + F ( t ) , (3)where d is the water depth in the absence of vegetation, β is the sensitivityof the water table to the presence of vegetation and F ( t ) describes the oscilla-tory variation of the water table. The choice F ( t ) = f cos ωt is considered in(Borgogno et al., 2012). In the present work we choose F ( t ) as a biharmonicforce with two widely differing frequencies: F ( t ) = f cos ωt + g cos Ω t, Ω ≫ ω. (4)The explicit time-dependent variation of the water depth can be natural dueto seasonal rainfall oscillations or man-induced perturbations (pumping froman aquifer). 7 a) V U ( V ) (b) V V cc Figure 1: (a) The potential U ( V ) of the system (1) for f = 0, g = 0. The values ofthe parameters are d = 0 . β = 0 . a = 26m − , d inf = 0 .
6m and d sup = 0 .
9m thevalues used in (Borgogno et al., 2012). (b) Variation of V cc with V (thick curve). Theequilibrium points are the intersections of the bisector (thin straight-line) with the V cc curve. The locations of the equilibrium points are marked by solid circles. The potential U ( V ) defined through d V / d t = − d U/ d V in the absenceof F ( t ) is depicted in Fig. 1a where d = 0 . β = 0 . a = 26m − , d inf = 0 .
6m and d sup = 0 .
9m the values of the parameters used in (Borgognoet al., 2012). U ( V ) is of a double-well form. The equilibrium states can beobtained by setting d V / d t = 0. V ∗ = 0, representing the unvegetated state,is an equilibrium point. The other equilibrium states correspond to V cc = V .In the plot between V cc versus V the intersection points of the line V cc = V with the curve of V cc are the equilibrium states. Figure 1b shows V cc versus V . The equilibrium states are V ∗ = 0, V ∗ u = 0 . V ∗ s = 0 . V ∗ u is the local maximum of U ( V ) and is an unstable state. V ∗ and V ∗ s are twolocal minima of U ( V ) and are stable states. When the periodic function F ( t )8s taken into consideration, then the effect of F ( t ) is to periodically modulatethe potential U ( V ). In the F ( t ) given by Eq. (4), we have assumed that Ω ≫ ω . In this casedue to the difference in time scales of the low-frequency oscillation f cos ωt and the high-frequency oscillation g cos Ω t , the solution of the system (1)consists of a slow variation of V ( t ) denoted by V slow ( t ) and a fast variation V fast ( t, Ω t ). We denote Q ω and Q Ω as the response amplitudes of V at thefrequencies ω and Ω respectively. A theoretical approach has been developedto obtain an analytical expression for Q for certain class of oscillators (Landaand McClintock, 2000; Blekhman and Landa, 2004). The threshold existingbetween the carrying capacity V cc and d in Eq. (2) makes it difficult to usethe theoretical approach to investigate the vibrational resonance. Therefore,we compute both Q ω and Q Ω from the numerical solution of the Eq. (1).From V ( t ) the sine and cosine components Q ω, s and Q ω, c are computed fromthe equations Q ω, s = 2 nT Z nT V ( t ) sin ωt d t, (5) Q ω, c = 2 nT Z nT V ( t ) cos ωt d t, (6)where T = 2 π/ω and n is taken as, say, 200. Then Q ω = q Q ω, s + Q ω, c . f. (7)Similarly, we can compute Q Ω from the numerical solution V ( t ).The value of f is significant in observing vibrational resonance in thesystem (1). For a fixed value of ω and g = 0, small amplitude oscillations9f V occur about the coexisting stable equilibrium points V ∗ and V ∗ s for | f | ≪
1. Then for fixed values of f with | f | ≪ ≫ ω , when g is varied both Q ω and Q Ω display one or more resonances depending uponthe values of the parameters of the system. For g > .
5, both Q ω and Q Ω ≈ < g < .
5. Figure 2 shows the dependence of Q ω and Q Ω for threefixed values of f with ω = 1 and Ω = 10. In all the cases a nonmonotonic f = 0 . f = 0 . f = 0 . (a) g Q ω f = 0 . f = 0 . f = 0 . (b) g Q Ω Figure 2: Response amplitudes (a) Q ω and (b) Q Ω of plant biomass V for three fixedvalues of the parameter f with ω = 1 and Ω = 10. The values of the other parameters inthe system (1) are as in Fig. 1. Q ω and Q Ω occurs. In Fig. 2 Q ω and Q Ω are ≈ g > . f = 0 .
08 as g increases from a small value, Q ω initially decreases thenincreases sharply, reaches a maximum at g = g VR1 = 0 .
069 and then sharplydecreases to a lower value. As g increases further, Q ω becomes maximum attwo other values of g , namely g = g VR2 = 0 .
126 and g = g VR3 = 0 . g of the high-frequency oscillation of the water table. For this reason theabove resonance phenomenon, that is the occurrence of a maximum of Q ω ,is termed as vibrational resonance . For f = 0 .
05 also three resonances occur.For both f = 0 .
05 and 0 .
08 the first resonance is the dominant and the valueof Q at the successive resonances decreases. On the other hand, the width ofthe successive bell-shaped resonance curves becomes wider and wider. Whenthe value of f increases the resonance peaks move towards lower values of g .For f values above a critical value, the first two resonance peaks disappear.Moreover, the value of Q at g = 0 is substantially enhanced with an increasein the values of f producing a resonance without tuning.In the nonlinear oscillators driven additively by a biharmonic force, Q Ω monotonically increases when g increases and there is no resonance-like vari-ation of it. We wish to remark that in the system (1), the biharmonic force isnot an additive force but it is in the expression for V cc . Therefore, one wishto know the response of the system (1) and its relative strength at the high-frequency Ω of the water table oscillation. For this purpose, we computedand presented the variation of Q Ω with g . Interestingly, Q Ω also displaysresonance. This is shown in Fig. 2b. The dependence of Q Ω on g is similar11o that of Q ω . For f = 0 .
05 and 0 .
08 there are two resonances with Q Ω andboth occur at the same values of g at which Q ω becomes maximum. Q Ω alsoshows resonance without tuning. Though the resonances associated with thetwo frequencies ω and Ω occur at the same value of the control parameter g , at resonance Q ω is much higher than Q Ω . We note that Q ω and Q Ω areproportional to the Fourier coefficients of the periodic terms with the fre-quencies ω and Ω respectively in the Fourier series of V ( t ). Since the Fouriercoefficients decay with increase in the frequency and because Ω ≫ ω the re-sponse amplitude Q Ω is much less than Q ω for each value of g in Fig. 2. It isnoteworthy to mention that at g = g VR an enhanced vegetation can be real-ized over two time intervals, one with the low-frequency ω and another withthe high-frequency Ω. These two frequencies need not be commensurable. V and V cc at and far from resonances The occurrence of resonance in the system (1) can be understood byanalyzing the influence of the control parameter g on the evolution of theplant biomass V and the carrying capacity V cc of the ecosystem. Figure 3shows V ( t ) and the carrying capacity V cc ( t ) of vegetation biomass for severalfixed values of g . In the absence of F ( t ) there are two stable dynamicalstates V ∗ and V ∗ s . The values of f and ω are chosen in such a way that inthe absence of a high-frequency oscillation of F ( t ) the variable V ( t ) oscillateseither about V ∗ or V ∗ s depending upon the initial condition and there is notransition between these two states. When g is varied from 0 then for verysmall values two oscillatory states coexist. The equilibrium states aboutwhich oscillation occurs are perturbed by F ( t ). For the entire range of valuesof g considered in Fig. 2 the evolution of V is periodic. V ( t ) is said to be12 a) g = 0 . (b) g = g VR1 = 0 . (c) g = 0 . (d) g = g VR2 = 0 . (e) g = 0 . (f) g = g VR3 = 0 . t V (g) g = 0 . (h) g = g VR1 = 0 . (i) g = 0 . (j) g = g VR2 = 0 . (k) g = 0 . (l) g = g VR3 = 0 . t V cc Figure 3: The time evolution of (a)-(f) V ( t ) and (g)-(l) V cc ( t ) of the system (1) at fewvalues of g with f = 0 . ω = 1 and Ω = 10. The values of the other parameters are asin Fig. 1. The dashed curves represent f cos ωt . For comparative purpose the amplitudeof f cos ωt is suitably multiplied by a factor. t ′ if V ( t + t ′ ) = V ( t ) for some finite and nonzero valueof t ′ after leaving initial transient evolution of it, say for example leaving V ( t ) for 0 < t < × T (= 2 π/ω ). The period of V ( t ) is found to be T = 2 π/ω , the period of F ( t ) given by Eq. (4). In Fig. 3a for g = 0 .
02 farbefore the first resonance, for the initial condition chosen near V ∗ s , thoughboth V ( t ) and f cos ωt are periodic with the same period T the forms ofboth of them are different. Over one drive cycle both V ( t ) and V cc ( t ) (seeFig. 3g) have more than one dominant maximum while F ( t ) has only one.There is no synchronization between V (as well as V cc ) and f cos ωt . For g = g VR1 = 0 .
069 in Figs. 3b and h the center of oscillation is shifted. V and V cc are synchronized with f cos ωt (except that there is a phase difference).This feature of V leads to the first resonance. The numerical analysis showsan absence of resonance at g = 0 .
069 for the vegetation dynamics aboutthe other low value stable state ( V ∗ ). The value of Q of the correspondingdynamics is very small.Next, at g = 0 .
07 the high-frequency oscillation of the water table inducesa transition of V about V ∗ s to V ∗ . When the vegetation biomass is close to V ∗ = 0, it is easy to note from Eqs. (2) and (3) that, V cc = 0 for most ofthe time over a drive cycle of F ( t ). Consequently, the dynamics is confinednear V ∗ . There is no transition between the two states V ∗ and V ∗ s . This isshown in Figs. 3c and i for g = 0 . V and V cc are quitesynchronized with f cos ωt , there is no resonance because V is trapped intothe unvegetated state. Because the center of oscillation of V is shifted fromthe vegetated state to the unvegetated state both Q ω and Q Ω make a suddenjump from a higher value to a smaller value at g = 0 .
07 (see Fig. 2). When14 increases further, the time intervals in which V cc = 0 decreases. In Fig. 3jcorresponding to g = 0 .
126 the total time over which V cc = 0 is ≈ T / V cc (indicated by solid circles connected by aline) varies sinusoidally and one can clearly notice synchronization between V cc and f cos ωt . A second resonance occurs at this value of g . In Fig. 3d V oscillates about the perturbed equilibrium state V ∗ u . For g = 0 .
23 rapidoscillations of V and V cc (Figs. 3e and k) take place and Q ω ≈
0. At g = 0 . Q ω much less than that of the first two resonancesoccurs. However, we can clearly see the synchronization between V and f cos ωt in (Fig. 3f).In this section so far we reported the results for ω = 1, Ω = 10 and f = 0 .
08. We numerically studied the occurrence of resonance for a widerange of fixed values of ω , Ω and f and thereby varying the control parameter g . Figures 4a and b show the variation of Q ω with g for ω ∈ [0 . , .
5] whereΩ = 10 and Ω ∈ [5 ,
15] where ω = 1 respectively. Three resonances occur fora wide range of values of ω and Ω. Figures 4c and d illustrate the influence ofthe parameter f on resonance for two sets of values of ω and Ω. In both thecases we observe one or more resonance for f ≪
1. The values of g at whichresonance occur and the corresponding value of Q vary with the parametersof the system.
3. Vibrational resonance in the two species model
In the previous section, we have described the vibrational resonance asso-ciated with the vegetation dynamics of a single species interacting with thewater table. The present section is devoted to a two phreatophyte species A ω ω g (a) Q ω Ω g (b) Ω=10, ω=1, Q ω f g (c) ω=1, Ω=10 Q ω f g (d) ω=0.5, Ω=5 f =0.08 f =0.08 Figure 4: Variation of Q ω as a function of (a) ω and g for Ω = 10, f = 0 .
08, (b) Ω and g for ω = 1 and f = 0 .
08, (c)-(d) f and g for two sets of fixed values of ω and Ω. Thevalues of the other parameters are as in Fig. 1. and B interacting with the water table. In the two species (denoted by A and B ) model proposed by (Ridolfi etal., 2006, 2007) the species A is assumed to be dominant over the species B .That is, in the absence of interactions with the water table, A tends to itsmaximum density while B tends to disappear. The values of A ( t ) and B ( t )16re normalized with respect to their maximum value. Frequently, the logisticlaw is chosen for the growth of A and B . The model system is thusd A d t = α A A ( V c A − A ) , (8)d B d t = α B B ( V c B − A − B ) , (9)where α A and α B are the coefficients determining the response rate of A and B respectively and V c A and V c B are the carrying capacities of the species A and B respectively. V c A and V c B depend on the depth d ( t ) of the water table.Introducing the change of variables λ = α B /α A and t = α A t ′ (which makestime a dimensionless quantity), dropping the prime in t ′ and assuming thatthe dynamics of B depends on A ( t − τ ) where τ is a time-delay, the Eqs. (8-9)become d A d t = A ( V c A − A ) , (10)d B d t = λB ( V c B − A ( t − τ ) − B ) . (11)In Eqs. (10-11) V c i = θ [ d − d min ,i ] · θ [ d max ,i − d ] , (12)where i = A, B , θ [ s ] is the Heaviside function, that is, θ [ s ] = 1 for s > s < d min ,i and d max ,i are the minimum and maximum water tabledepths tolerated respectively by the species i with d min ,B < d min ,A < d max ,B (species A needs a deeper aquifer than species B ) and d ( t ) = d + β A A + β B B + f cos ωt + g cos Ω t, (13)where d is the water table depth in the absence of vegetation ( A = B =0) and β A and β B are coefficients that weight the control exerted by thevegetation on the water table depth d .17n the above two species model one of the species is assumed to be sub-dominant. Further, the feedback of A on the evolution of B is taken as atime-delayed term. For a variety of phreatophytes these assumptions can berealized. Dominant species A can be regarded as plants with tap-roots ableto penetrate through relatively deeper than those of the species B . In theliterature of plants, approximate maximum lengths of tap-roots of phreato-phytes are reported. Some of the plants with deep penetrating tap-roots withlength more than 15m are mequite, camelthorns, grease wood, and purplemedic. Plants such as black grease wood and banksia (173 species) havetap-roots with length in the range of 5-10m. Examples of plants with tap-roots of short length about 1-5m are saguaro, creosote bush, ocotillo brittlebush, sagebrush, alder and chamisa. One can identify appropriate speciesof types A and B . In this connection we wish to cite that the experimentalanalysis carried out on holm oaks and cork oaks indicated that the higherwater status leading to more effective drought avoidance of former is dueto their deeper root systems compared to the latter (David et al., 2007).A field experiment was performed on the two phreatophytic plant speciesAlhagi sparsifolia (camelthorns) and Karelinia caspia occurring around theriver Oasis at the southern fringe of the Taklamakan desert (Vonlanthen etal., 2010). Both the species occur at sites with distances to the ground watertable upto 12m while only Alhagi sparsifolia occurs at distances upto 17m.The motivation for introducing the time-delay term A ( t − τ ) in the system(10-11) is to take into account the fact that the changes in the populationof a species, generally, will not have immediate effect on the growth of ownpopulation and on the interacting species. The effect will be realized after18 time-lag. The effect of time-delay has been studied in population modelsand vegetation dynamics (Kuang, 1993; Wang et al., 2011). Reports on theanalysis of influence of various factors on the growth of vegetation dynam-ics based on AVHRR (Advanced Very High Resolution Radiometer) imagesindicate that the time-lag can be few days to few months (Richard and Poc-card, 1998; Li et al., 2002; Wang et al., 2006; Farajzadeh et al., 2011). Areasonable value of time-delay can be of the order of 1 /α B where α B is thecoefficient determining the response rate of the species B . For our numerical study we fix d = 1m, d min ,A = 1 . d max ,A = 2 . d min ,B = 0 . d max ,B = 2m, β A = 0 . β B = 0 . α A = α B = 1d − , ω = 0 . f = 0 and g = 0 the system (10-11) hastwo equilibrium states. ( A ∗ , B ∗ ) = (1 ,
0) is stable while ( A ∗ , B ∗ ) = (0 ,
1) isunstable. ( A ∗ , B ∗ ) = (1 ,
0) remains as a stable equilibrium point for f < . ω = 0 . g = 0 and τ = 1. The system exhibits excitable dynamics for f > . f less than 0 .
01 so that in the absence of high-frequency oscillation of water table the system is in the stable equilibriumstate. It is important to investigate the response of the system for a widerange of values of the control parameters f , ω , g , Ω and τ . Because sucha study in the five parameters space is time consuming we restrict to ( g, τ )parameters space. We studied the response of the system for a wide rangeof values of g for several fixed values of the parameters of the system. For g > . A and B are ≈
0. Therefore, we consider the range of g as [0 , . g, τ ) parameter space first we discuss thepresence of resonance for fixed values of f , ω , Ω and τ thereby varying theparameter g .Figure 5 presents the numerically computed response amplitudes Q A ( ω )and Q B ( ω ) as a function of g for three fixed values of f with τ = 1. Thereare few interesting results: (a) f = 0 . f = 0 . f = 0 . g Q A (b) f = 0 . f = 0 . f = 0 . g Q B Figure 5: Response amplitudes Q A and Q B of species A and B versus the parameter g forthree fixed values of f . The values of the other parameters are d = 1m, d min ,A = 1 . d max ,A = 2 . d min ,B = 0 . d max ,B = 2m, β A = 0 . β B = 0 . α A = α B = 1d − , ω = 0 .
5, Ω = 5 and τ = 1. The response amplitudes of the species A and B display similar varia-tion, however, Q A > Q B , that is, the species B is subdominant. • An interesting result is that even though species A is dominant, when A exhibits resonance the species B does not disappear but also displaysa resonance and, moreover, in Fig. 5 we notice that Q B ≈ Q A / • Multiple resonance occurs. Q A and Q B are not maximum at discretevalues of g , but they are almost constant over a range of values of g . • Q A (and Q B ) values at successive resonances are not equal. In thesingle species model also response amplitude values are not same atsuccessive resonances. We wish to point out that in bistable systemsdriven additively by a biharmonic force the response amplitude at suc-cessive resonances are found to be the same (Landa and McClintock,2000; Rajasekar et al., 2010). • Q A (and Q B ) values at resonances decrease by increasing the value of f , while the width of the resonance interval increases.To understand the occurrence of a multiple resonance and almost plateauregions of resonance profile, we consider the nature of the evolution of thesystem. The system (10-11) is a system of two-coupled first-order nonlineardifferential equations driven by a periodic force. Such a system is capableof exhibiting different types of nonlinear dynamics including chaotic dynam-ics (a nonperiodic and bounded evolution of a system with high sensitivedependence on initial conditions). In the system (10-11) in the absence of21ime-delay and high-frequency oscillation of water table chaotic dynamics isfound to occur when the amplitude of the low-frequency oscillation of thewater table is varied (Ridolfi et al., 2007). However, in the system (10-11)for the parametric choices considered in the present work any route to chaoticdynamics is not observed.In Fig. 5 the variation of Q A and Q B is shown for g ∈ [0 , .
4] only. For g > .
4, as mentioned earlier, we found Q A and Q B ≈ < g < .
4. In this interval of g eitherstable equilibrium state or periodic variation of A and B is found dependingupon the value of g . To identify the period of A ( t ) we collect the valuesof A ( t ) at t = nT where n = 1 , , · · · , m (= 500), T = 2 π/ω and designatethem as A n . If A ( t ) is periodic with period T then A = A = · · · = A m .For a period-2 T variation of A ( t ) we observe A = A = · · · = A m − = a , A = A = · · · = A m = a and a = a . Similarly, we can define higherperiods of A and identify the periodicity of A . In Fig. 6 we have plotted theperiod of A together with Q A for f = 0 . A is a complicatedfunction of g .Interestingly, the period remains the same in the regions of g where Q isalmost a constant. Different plateau regions correspond to different constantperiods of A . For g < .
007 we observe (
A, B ) → (1 ,
0) as t increases. Thespecies biomass A attains its maximum while that of the species B failedto survive. Even though the water table depth d ( t ) varies periodically, theevolution of A and B is not oscillatory but approaches the equilibrium statefor g < .
007 as shown in Figs. 7a and e for g = 0 . d ( t ) does not strictly appear as an additive driving22 A g P e r i o d o f A (a)(b) Figure 6: Plots of (a) the response amplitude Q A and (b) the period of A (in units of T )versus g for f = 0 . g with largevalues of Q A are marked by the numbers 1, 2 and 3. The values of the other parametersin the system (10-11) are as in Fig. 5. force in (10-11) but its influence is on V c A and V c B . Equations (10-11) can beregarded as a parametrically driven system. Moreover, V c A and V c B are notsinusoidally varying function of time. They take the values 0 or 1 dependingupon the values of the various parameters of the system and the value of t .The most dominant resonance interval is g ∈ [0 . , . g the period of A is T (see Fig. 6b). Figures 7c and g show the evolution of A and B with time for g = 0 .
2. We can clearly notice the best synchronizationof A and B with the external drive f cos ωt . The next dominant resonanceinterval is g ∈ [0 . , . A in23 a) g = 0 . (b) g = 0 . (c) g = 0 . (d) g = 0 . t A (e) g = 0 . (f) g = 0 . (g) g = 0 . (h) g = 0 . t B Figure 7: Temporal behaviour of (a)-(d) A and (e)-(h) B for four fixed values of g with f = 0 . f cos ωt and its amplitude is suitably magnified and its center ofoscillation is shifted for illustrative purpose. this interval of g is 2 T . In Figs. 7b and 7f for g = 0 .
065 we observe pulse-like solution with period-2 T . Period-3 T oscillation of A and B occurs in theinterval g ∈ [0 . , . g > .
24 the evolution of A and B exhibits a rapid oscillation (asshown in Figs. 7d and h for g = 0 . Q A and Q B decays tozero implying degrading of the response of the biomasses A and B with24ncrease in the value of g . Outside the three dominant resonance regions,we observe few small intervals of g where Q becomes almost constant butrelatively very small due to higher periodicity of variation of A and B . InFig. 7 we notice that as g increases from a small value the evolution of A and B undergoes a transition from equilibrium state → pulse-like solution → rapid oscillatory solution. Further, when A becomes maximum (minimum) B becomes minimum (maximum). From the above, we point out the followingfeatures: • Q is very small if the species biomass remains a constant (see Figs. 7aand e) or periodic with large period or it oscillates rapidly (see Figs. 7dand h). • Q is larger, for example the regions 1, 2 and 3 in Fig. 6a, when theperiod of the pulse-like evolution of the biomass species is lower. • Q remains almost constant over an interval of g (regions 1, 2 and 3 ofFig. 6a) if the period of A remains the same.It is noteworthy to compare the mechanisms of vibrational resonancein the FitzHugh–Nagumo (FN) equation, a well studied excitable system,and the system (10-11). In the FN equation when the amplitude of thehigh-frequency force varies, a resonance takes place when the waiting time( T w ), the time the system spends around the equilibrium point between twoconsecutive firing, is T / T w ≈ T /
2. Inthe system (10-11), Q is inversely proportional to the period of evolution ofbiomasses. 25 .3. Effect of a time-delay on the resonance So far, we have focused our analysis for a specific value of the time-delay τ (= 1). Now, we present the effect of time-delay on the resonance.First, in the parameter space ( g, τ ), we identify the regions where Q ( τ, g ) >Q (0 , g ) for both species. We divide the ( g, τ ) parameter space in the interval g ∈ [0 , .
4] and τ ∈ [0 ,
12] into 100 ×
100 grid points. We collect the gridpoints for which Q ( τ, g ) > Q (0 , g ). The result is the Fig. 8. Both Figs. 8aand b corresponding to the species A and B respectively are almost similarexcept near g = 0 (far before resonance) and near g = 0 . Q A and Q B are very small. Thetime-delay τ has a strong influence on the response amplitudes of A and B .Next, Fig. 9 features the colour-contour plots of the dependence of Q A and Q B on g and τ . Multiple resonance is found to occur for a wide range offixed values of τ . Moreover, Fig. 9 clearly demonstrates the strong influenceof τ on the variation of Q . For fixed values of g the response amplitudes Q A and Q B are not periodic with τ . Figure 10 depicts the dependence of Q A on the time-delay τ for g = 0 . Q A (as well as Q B ) exhibits a sequence ofresonance intervals. The period of A (as well as B ) in the first resonanceinterval is 2 T . The periods of A in the other consecutive intervals are 3 T ,4 T , · · · respectively. That is, the system exhibits a resonance sequence withperiod adding dynamics. The resonance intervals with period of A being even(odd) integer multiples of T are marked by open (solid) circles in Fig. 10. Thevalues of Q A of these two sequence of resonance intervals decrease rapidlywith τ . Further, we observe that Q A ( τ ) does not varies periodically with τ , that is Q A ( τ + α ) = Q A ( τ ) for some finite nonzero value of α . However,26 a) g τ (b) g τ Figure 8: Regions (marked by black colour) where (a) Q A ( τ, g ) > Q A ( τ = 0 , g ) and (b) Q B ( τ, g ) > Q B ( τ = 0 , g ) for the system (10-11). The values of the parameters are as inFig. 5. the resonance intervals marked by solid circles and open circles occur at aregular interval of delay-time ≈ T . Similar dependence of Q on τ is foundto occur for other fixed values of g . We note that in nonlinear oscillatorswith time-delayed feedback and driven additively by a biharmonic force Q isshown to vary periodically (Yang and Liu, 2010; Jeevarathinam et al., 2011).In this section we presented our analysis on the system (10-11) for ω = 0 . ω and Ω are observedfor a wide range of values of ω and Ω. The number of resonances, the values27 (a) g τ (b) g τ Figure 9: (Colour in online) Colour-coded dependence of Q A and Q B of the system (10-11)on the parameters τ and g for f = 0 . of Q at resonances and the values of g at which resonances occur depend onthe values of the other parameters of the system.
4. Conclusion
Several studies have reported the occurrence of vibrational resonance in-duced by a two-frequency periodic force in physical and biological nonlinearsystems. The present work is devoted to the analysis of the effect of a bi-harmonic type variation of the water table depth in two phreatophyte plant28 = 0 . τ Q A Figure 10: Q A of the system (10-11) as a function of time-delay τ for g = 0 .
1. The valuesof other parameters are as in Fig. 5. The period of A in units of T (= 2 π/ω ) is markedfor first few resonance intervals. The resonance intervals with period of A equal to even(odd) integer multiples of T are marked by open (solid) circles. ecosystems. We have considered a single species and a two species model sys-tems. The former has bistability, while the later is an excitable system. Ourstudy shows how a very simple deterministic periodic variation of the waterdepth is able to give rise to a great increase in the response of vegetationdynamics of single species and two species ecosystems.In both overdamped and underdamped nonlinear oscillators exhibitingmultiple vibrational resonance (Landa and McClintock, 2000; Baltanas etal., 2003; Jeyakumari et al., 2009; Yang and Liu, 2010; Jeevarathinam et al.,2011), at all resonances the long time motion of the systems is found to beperiodic with period T of the low-frequency force when the ratio Ω /ω is aninteger. In the single species system (1) the period of evolution is T at allthe resonances, however, the value of Q at successive resonances decreases.In the two species model, the period of the system is different at resonances.Further, the response amplitude Q is found to be inversely proportional to29he period of the evolution of the species biomass. Analysis of vibrationalresonance in systems described by different kinds of evolution equations canlead to a deeper understanding of the phenomenon.We believe that analysis of nonlinear phenomena such as chaos, stochas-tic and vibrational resonances, in the theoretical models of vegetation-watertable interactions may motivate the experimentalists to perform experimentswith controlled variations of the water table depth in a small scale. Suchstudies not only would explore the ways of enhancing the biomass response,but also help to improve the theoretical models based on experimental ob-servations. Acknowledgments
CJ acknowledges the support from University Grants Commission, Indiain the form of UGC-Meritorious Fellowship. One of us (SR) would like tothank M. Sundararaman for his helpful discussions. Financial support fromthe Spanish Ministry of Science and Innovation under Project No. FIS2009-09898 is acknowledged by MAFS. We gratefully thank anonymous refereesfor their constructive comments and suggestions on our earlier draft whichimproved the quality and presentation of the paper.
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