Restoration and recovery of damaged eco-epidemiological systems: Application to the Salton Sea, California, USA

Mathematical Biosciences 242 (2013) 172–187

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Mathematical Biosciences

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Restoration and recovery of damaged eco-epidemiological systems: Applicationto the Salton Sea, California, USA

Ranjit Kumar Upadhyay a,⇑, S.N. Raw a, P. Roy a, Vikas Rai b

a Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand 826 004, Indiab Department of Mathematics, Faculty of Science, Jazan University, Jazan, P.O. Box 114, Saudi Arabia

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 August 2012Received in revised form 8 January 2013Accepted 11 January 2013Available online 8 February 2013

Keywords:Salton lakeAgricultural runoffEco-epidemiological modelAvian botulismFishesBiodiversity

0025-5564/$ - see front matter � 2013 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.mbs.2013.01.002

⇑ Corresponding author. Tel.: +91 326 2235482; faxE-mail address: [email protected] (R.K. Upad

In this paper, we have proposed and analysed a mathematical model to figure out possible ways to rescuea damaged eco-epidemiological system. Our strategy of rescue is based on the realization of the fact thatchaotic dynamics often associated with excursions of system dynamics to extinction–sized densities.Chaotic dynamics of the model is depicted by 2D scans, bifurcation analysis, largest Lyapunov exponentand basin boundary calculations. 2D scan results show that l, the total death rate of infected prey shouldbe brought down in order to avoid chaotic dynamics. We have carried out linear and nonlinear stabilityanalysis and obtained Hopf-bifurcation and persistence criteria of the proposed model system.

The other outcome of this study is a suggestion which involves removal of infected fishes at regularinterval of time. The estimation of timing and periodicity of the removal exercises would be decidedby the nature of infection more than anything else. If this suggestion is carefully worked out and imple-mented, it would be most effective in restoring the health of the ecosystem which has immense ecolog-ical, economic and aesthetic potential. We discuss the implications of this result to Salton Sea, California,USA. The restoration of the Salton Sea provides a perspective for conservation and management strategy.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Eco-epidemiological modeling helps us to understand howchemical and physical factors interact to shape the environmentand how both the factors contribute to the recovery, restorationand health of the ecosystem. Changes in the chemical and physicalnature of a site can impact the fish, algae and invertebrate popula-tion which live in that environment. Anderson and May [2] werefirst who merged the two fields (Ecology and Epidemiology) andformulated a predator–prey model where prey species were in-fected by some disease. Predator may even prevent successfulinvasion of parasites into host population. In most theoreticalstudies of host-parasite-predation interaction, predator behavioris simplified and isolated from an ecosystem. Packer et al. [29]studied eco-epidemiological models for micro-parasitic andmacro-parasitic infections with constant predator and showed thatpredator removal was more likely to be harmful when the parasitewas highly virulent. Han et al. [17] analyzed four eco-epidemiolog-ical models (SI and SIR types) with standard and mass action inci-dences. Different thresholds were identified and global stabilityresults were proved. It was shown that disease might persist inthe prey and predator populations if the basic reproduction forprey population is above some threshold and feeding efficiency

ll rights reserved.

: +91 326 2296563.hyay).

of the predator population is significantly high. Chen and Jiang[8] studied the stability and direction of bifurcating periodic solu-tions performed by normal form theory and center manifoldargument.

Bairagi et al. [4] studied an eco-epidemiological model with theassumption that both the infected and non-infected prey are sub-ject to combined harvesting. Haque and Venturino [18] modifiedthe classical Holling–Tanner model allowing a disease to spreadamong the prey species. They showed that introduction of a dis-ease in a pure demographic model can destabilize the otherwisestable system and sometimes the predator might act as a systempreserver. Mukherjee [26] proposed and analyzed a three specieseco-epidemic model. He considered Holling type II predator func-tional response and investigated the condition for limit cycle toarise by Hopf bifurcation and also the criterion for the existenceof Hopf-type small periodic oscillation. Bhattacharyya andMukhopadhyay [5] concentrated on the mathematical aspects ofdisease dynamics with reference to a model at the interface of ecol-ogy and epidemiology. Recently, Niu et al. [27] studied the asymp-totic behavior of a nonautonomous eco-epidemic model withdisease in the prey. They established the necessary and sufficientconditions on the permanence and extinction of the infective preyand the sufficient conditions on the global attraction of the model.Sarwardi et al. [33] modified the Leslie-Gower type II model tointroduce a contagious disease in the predator population and as-sumed that disease cannot propagate to the prey. Chattopadhyay

R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187 173

et al. [7] showed that there always exists a Hopf-bifurcation forincreasing transmission rate in a classical predator–prey systemwith infection in prey population. Fenton and Rands [12] studiedtwo eco-epidemiological models where prey was infected eitherby micro-parasite or by macro-parasite. They showed that preymanipulation by parasites can greatly alter the quantitativedynamics of the community, potentially resulting in high ampli-tude oscillations in abundance, but the precise outcome of theinteraction depends on both, the form of manipulation and the nat-ure of the predator’s functional response.

Chattopadhyay and Bairagi [6] proposed and studied an eco-epidemiological model (SI type) with the assumption that predatorconsumes infected prey only. They observed that the system is sta-ble around the positive interior equilibrium if the search rate levelof predator is low, but the instability sets in when search rate levelincreases. Upadhyay et al. [39] modified the eco-epidemiologicalmodel of Chattopadhyay and Bairagi [6] by taking into accountthe bilinear mass action incidence rate and studied it numericallyfor larger ranges of different system parameters. Their simulationresults suggest that an eco-epidemiological system may displaydistinct dynamical behaviors starting from stable focus/stable limitcycle to chaos. Recently, Mandal and Banerjee [24] revisited themodel studied by Upadhyay et al. [39] and derived the conditionfor Hopf-bifurcation which is responsible for initiation of smallamplitude periodic solution around co-existing steady-state. Thedevelopment of our eco-epidemiological model system depictsthat the present model system is sensitive to several parametersin contrast to the model studied by Mandal and Banerjee [24].We have assumed that predators are not smart enough to distin-guish between infected and healthy prey and predator’s responseto the infected prey and susceptible prey follows Holling type IVand Holling type II response functions, respectively. Other pub-lished models in the literature consider same type of functional re-sponses for both the susceptible and infected prey. Holling type IVfunction for the predation of infected prey is not considered by anyauthor till date. The present model is a new hybrid model whichincorporates both Holling type II and IV functional responses.

This manuscript explores chaotic predator–prey dynamics inthe marine system (e.g. Salton Sea) and its implications for conser-vation and management of this ecosystem. We have also tried toexplain the unusual deaths of fish and fish eating birds in SaltonSea using the bifurcation analysis results. However, further inves-tigations are required to answer the following issues. It is knownthat Tilapia fish has stunning reproduction rate [10]. Is this higherreproduction rate responsible for the instability of the system? Didthe system witness a chaotic dynamics that caused massive fishand bird mortality events? To predict the real life situation fromthe model system, it is necessary to test the model under differentinitial conditions and study its dynamical behavior over widerranges of parameter values. We have done extensive numericalstudies of the model system to determine the regions in theparameter spaces, which support different dynamical behavior ofthe system including extinction of species.

The Salton Sea, which is located in the southeast desert of Califor-nia, came into limelight due to deaths of fish and fish eating birds ona massive scale (8 million fish died in a single day on 12th August1999 [25]). In Salton Sea, fish population, infected from both the vib-rio and botulism, swim so slowly that they become an easy catch forpiscivorous birds. Predatory birds are infected in turn when theyconsume heavily infected fish [14,21,22]. The quantity of the agri-cultural drainage sustains the Sea and the quality of the drainageis responsible for its problems. The Salton Sea is a terminal lake.The only outflow of its water is through evaporation. As water evap-orates, salts, selenium and other contaminants are concentrated inthe sea and its sediments. In the first four months of 1998 alone,17,000 birds from 70 species died from a variety of diseases [9].

Massive fish die – offs occur due to eutrophic conditions prevailingin the sea. The increasing salinity of the Salton Sea jeopardizes thefuture existence of fish in the sea [32]. Since the early 1990s fre-quent large scale die–offs of fishes and birds have been a cause ofmajor concern. Low oxygen levels and disease causing agents suchas bacterial infections, algal toxins and parasites could be the causeof massive die – offs [11]. Botulism outbreak at Salton Sea in 1996killed nearly 15% of American white pelican, Pelecanus erythrorhyn-chos. Occurrence of avian botulism is largely controlled by environ-mental factors. A neurotoxin produced by the bacterium Clostridiumbotulinum causes avaian botulism. It can also be caused by ‘‘toxico-infections’’, when botulinum toxin producing bacteria colonize theintestinal tract of an individual. Most of the botulism outbreaksare caused by type C toxin. The main outcome of the above studieswas – harvesting the infected Tilapia population is needed for eco-logical sustainability of the Salton Sea.

The paper is organized as follows. In Section 2, we discuss a neweco-epidemiological model system and its dissipativeness. Sec-tion 3 deals with analysis of the model system. Linear and nonlin-ear stability analysis of the model system, Hopf-bifurcation andpersistence criteria of the model system are presented in this sec-tion. Section 4 depicts the numerical simulation results. We estab-lish its application to the Salton Sea in Section 5. Finally, discussionon the results is presented in the last Section.

2. The development of eco-epidemiological model system

We consider an eco-epidemiological model which consists oftwo constituent populations; i.e., prey population (Tilapia fish)and predator population (Pelicans). The prey population densityis denoted by N(t) and the predator population density by P(t) attime t. We impose the following assumptions to formulate the dif-ferential equations which describe the model system.

Assumption 1. In the absence of bacterial infection, the fishpopulation grows according to a logistic fashion with carryingcapacity K(K e R+) and birth rate constant r(r e R+) such that

dNdt¼ rN 1� N

K

� �: ð1Þ

Assumption 2. In the presence of bacterial infection, we assumethat the total fish population N is divided into two classes, namely,susceptible fish population, denoted by S, and infected fish popula-tion, denoted by I. Therefore, at any time t, the total number of fishpopulation isNðtÞ ¼ SðtÞ þ IðtÞ: ð2Þ

Assumption 3. We assume that only susceptible fish population, S,is capable of reproducing with logistic law (Eq. (1)) and the infec-tive fish population, I, does not reproduce. However, the infectivefish, I, still contributes with S to population growth towards thecarrying capacity.

Assumption 4. The mode of disease transmission follows the sim-ple law of mass action. The disease is spread among the prey pop-ulation only and the disease is not genetically inherited. Theinfected population does not recover or become immune. There-fore, the evolution equation for the susceptible fish population, S,according to Eq. (1) and assumptions (3) and (4), can be written as

dSdt¼ rS 1� Sþ I

K

� �� kSI; ð3Þ

where kðk 2 RþÞ is the rate of transmission (or force of infection).

174 R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187

Assumptions 5. Since prey population are infected by a (lethal)disease: infected preys are weakened and become easier to catch.Also they are present in the Salton Sea in considerable number,so we assume that predator’s functional response to the infectiveprey follows Holling type IV functional response [3,40,41] andincluded in the predator’s growth equation with a positive sign.While susceptible preys easily escape and predation becomes diffi-cult. We also assume that predator’s functional response to thesusceptible prey follows Holling type II predation [20,39] formwhich is also included in the predator’s growth equation with apositive sign. We write down following set of differential equationswhich describes the present model:

dSdt¼ rS 1� Sþ I

K

� �� kSI � h1SP

Sþ d¼ Sg1ðS; I; PÞ; ð4aÞ

dIdt¼ kSI � m1IP

I2=iþ I þ a� lI ¼ Ig2ðS; I; PÞ; ð4bÞ

dPdt¼ h2SP

Sþ dþ m2IP

I2=iþ I þ a� dP ¼ Pg3ðS; I; PÞ; ð4cÞ

with Sð0Þ > 0; Ið0Þ > 0; Pð0Þ > 0.All the parameters r;K; k; h1; h2; d;m1;m2; i; a;l; and d are posi-

tive constant. r;K and k represent intrinsic birth rate, environ-mental carrying capacity and transmission rate or the force ofinception respectively. h1;m1 are the search rates, h2 represents theconversion factor, m2 represents the conversion factor of infectedTilapia to Pelicans, d measures the extent to which the environ-ment provides protection to susceptible Tilapia, i a direct measureof pelican bird’s immunity from or tolerance of infected Tilapia. a isthe half saturation constant, l = l1 + l2, total death rate of infectedprey where l1 is the natural death rates of infected prey popula-tion and l2 represents the rate of death of infected prey due toalgal bloom. d = d1 + d2 is the total death of predator population.d1ðd1 2 RþÞ represents the natural death rate of predator popula-tion and d2ðd2 2 RþÞ represents the rate of death due to predationof infected prey.

A brief description about variables and parameters used in themodel system (4) is presented below in Table 1.

Clearly the model system (4) has twelve parameters in all.Obviously the interaction functions giði ¼ 1;2;3Þ of the modelsystem (4) are continuous and have continuous partial derivatives

Table 1Notations used to denote variables and parameters.

Variable/parameter Units Descriptio

S Number per unit area SusceptibI Number per unit area Infected TP Number per unit area Pelicans pr Per day Intrinsic bK Number per unit area Environmk Per day Transmish1 Per day Search rah2 Per day Conversiod Number per unit area Measuresa Number per unit area Half satum1 Per day Search ram2 Per day Conversiolð¼ l1 þ l2Þ Per day Total deal1 Per day Natural dl2 Per day Rate of dei Per day A direct mdð¼ d1 þ d2Þ Per day Total dead1 Per day Natural dd2 Per day Rate of de

on R3þ ¼ fðS; I; PÞ 2 R3 : S > 0; I > 0; P > 0g. Therefore the solution

of the model system (4) with non-negative initial condition existsand is unique, as the solution of system (4) initiating in the non-negative octant is bounded. The following theorem establishes thedissipativeness of model system (4).

Theorem 1. If the following condition

m1

m2P

h1

h2; ð5Þ

holds, then model system (4) is dissipative.

Proof. We define a function

xðtÞ ¼ SðtÞ þ IðtÞ þ h1

h2PðtÞ ð6Þ

The time derivative of above along the solutions of (4) is

dxdt¼ rS 1� Sþ I

K

� �� lI � m1 �m2

h1

h2

� �IP

ðI2=iþ I þ aÞ� dh1

h2P;

6 rS 1� Sþ IK

� �� m1 �m2

h1

h2

� �IP

ðI2=iþ I þ aÞ

�minðl; dÞ I þ h1

h2P

� �:

Hence it is easy to verify that under the condition (5) weobtain

dxdt6 rS 1� Sþ I

K

� ��minðl; dÞ I þ h1

h2P

� �;

dxdtþ gx 6 rS 1� Sþ I

K

� �þ gS;

dxdtþ gx 6 ðr þ gÞK; ðsince SðtÞ þ IðtÞ 6 KÞ

(since SðtÞ þ IðtÞ 6 K) where g = min (l,d). Applying the com-parison lemma for t P ~T P 0. Then

xðtÞ 6 Kðr þ gÞg

� Kðr þ gÞg

�xð~TÞ� �

e�gðt�~TÞ:

n

le Tilapia populationilapia populationopulationirth rate constant of Tilapiaental carrying capacity

sion ratete of the susceptible Tilapian factor of susceptible Tilapia to Pelicansthe extent to which the environment provides protection to susceptible Tilapia

ration constant in the absence of any inhibitory effectte of the infected Tilapian factor of infected Tilapia to Pelicans

th rate of infected Tilapia from causes other than predationeath rate of infected Tilapiaath of infected Tilapia due to algal bloomeasure of pelican bird’s immunity from or tolerance of infected Tilapia

th rate of Pelicanseath rate of Pelicansath of Pelicans due to predation of infected Tilapia

Fig. 1. Attractors set for different value of a and keeping other parameters fixed as in Eq. (19): (a) strange attractor for a = 16, (b) period-4 for a = 16.7, (c) period-2 for a = 18,(d) period-1 for a = 18.3. (e) time series for the infected population for figure in (a).

R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187 175

Then for ~T ¼ 0 we have

xðtÞ 6 Kðr þ gÞg

� Kðr þ gÞg

�xð0Þ� �

e�gt :

For large value of t, we have

xðtÞ 6 Kð1þ rgÞ 8t P 0: ð7Þ

Thus xðtÞ ¼ SðtÞ þ IðtÞ þ h1h2

PðtÞ 6 Kð1þ rgÞ, and then all species

are uniformly bounded for any initial value in R3þ.

Note that, for a biologically realistic model system (4) has to bedissipative (i.e., all population are uniformly limited in time by theirenvironments). Therefore, according to the above theorem weassume that there exists ðg1;g2;g3Þ > 0 such that XðS0; I0; P0Þ� R3

þ ¼ fðS; I; PÞ : 0 < S 6 g1; 0 < I 6 g2; 0 < P 6 g3g for allðS0; I0; P0Þ > 0; where XðS0; I0; P0Þ is the omega limit set of the orbitinitiating at ðS0; I0; P0Þ. Thus, the model system (4) is dissipative. h

3. Analysis of the model system

There are two basic processes in the present model system: (1)infectious disease process and (2) predation process. We will

176 R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187

design two subsystems with these processes which decide thedynamics of the original model is oscillatory or chaotic dependingon coupling of these two subsystems is weak or strong. If both thesubsystems are in oscillatory mode, they can couple together togive rise to chaotic dynamics provided two subsystems are weaklycoupled. If the coupling of the two subsystems is strong, it wouldresult in oscillatory dynamics. The coupling mechanism in thepresent case is Holling type IV response function. In the designedmodel system, the parameter values are chosen on the basis of bio-logical principles and correspond to the quantitative measures ofattributes of the susceptible Tilapia fish and infected Tilapia-Peli-can bird population. An ecosystem is considered to in sound healthwhen all the constituent species coexist on a stable limit cycleattractor i.e., perform oscillatory motion. Chaotic dynamics resultsfrom a change in a critical parameter of the system. The presence ofchaotic dynamics in a system (cf. Fig. 1(e) for excursions to extinc-tion–sized densities) means that the system is susceptible to exog-enous stochastic influences and species loss may result in certaincircumstances. This explains the origin and biological significanceof the oscillatory and chaotic dynamics exhibited by the modelsystem.

3.1. Design of two subsystems

The first subsystem is obtained by assuming the absence ofpredator and with the disease process in the following form:

dSdt¼ S r 1� Sþ I

K

� �� kI

� �¼ SFðS; IÞ; ð8aÞ

dIdt¼ IðkS� lÞ ¼ IGðS; IÞ: ð8bÞ

However, the second subsystem is obtained in the absence ofthe infected prey and with the predation process which takes theform:

dSdt¼ S r 1� S

K

� �� h1P

Sþ d

� �¼ SFðS; PÞ ð9aÞ

dPdt¼ P

h2SSþ d

� d

� �¼ PGðS; PÞ ð9bÞ

Further, the linear stability analysis of the subsystems (8) and(9) gives the following results. There are three non-negative equi-librium points of subsystem (8).

(i) The point E00 = (0,0) always exists with eigen values r and�l and is a saddle point.

(ii) The point E10 = (K,0) always exists with eigen valuesð�lþ KkÞ and �r. It is saddle point for K > l=k; and islocally asymptotically stable point for K < l=k.

(iii) The positive equilibrium E20 ¼ ð�S;�IÞ ¼ ðl=k; rðkK � lÞ=kðkK þ rÞÞ exists under the condition K > l=k and it is locallyasymptotically stable. The eigenvalues are f�lr � ffiffiffiffiffiffilr

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4Kkðl� KkÞ þ lr

pg=2Kk.

Also there are three non-negative equilibrium points of the sub-system (9).

(i) The point �E00 ¼ ð0;0Þ always exists with eigen values �d; rand is a saddle point.

(ii) The point �E10 ¼ ðK;0Þ always exists with eigen values�dþ ðKh2Þ=ðK þ dÞ;�r and is a saddle point forK > dd=ðh2 � dÞ with h2 > d. It is locally asymptotically sta-ble for K < dd=ðh2 � dÞ with h2 > d.

(iii) The positive equilibrium

�E20 ¼ ð��S; ��PÞ¼ ðdd=ðh2 � dÞ; rh2dðKðh2 � dÞ � ddÞ=h1Kðh2 � dÞ2Þ

exists under the Kolmogorov condition K

> dd=ðh2 � dÞ; h2 > d;

and it is locally asymptotically stable iff2dd=Kðh2 � dÞ þ d=K � 1g > 0.

3.2. Linear stability analysis and Hopf-bifurcation

The model system (4) has the following equilibrium points:

(i) The trivial equilibrium point E0 ¼ ð0;0;0Þ always exists.(ii) The axial equilibrium point E1 ¼ ðK;0;0Þ exists on the

boundary of the first octant.(iii) The predator-free equilibrium point E2 ¼ ð�S;�I;0Þ is the pla-

ner equilibrium point on the SI-plane, where �S ¼ lk ;

�I ¼ rðkK�lÞkðrþkKÞ is positive if kK > l, same as given in E20.

(iv) The disease-free equilibrium point E3 ¼ ð��S;0; ��PÞ is the planerequilibrium point on the SP-plane, where ��S ¼ dd

ðh2�dÞ ;��P ¼ rdh2fKh2�dðKþdÞg

Kh1ðh2�dÞ2is positive if h2 > d;K > dd

ðh2�dÞ, same as givenin �E20.

(v) The nontrivial equilibrium E4 ¼ ðS�; I�; P�Þ exists in the int. R3þ

if and only if there is a positive solution to the following setof equation:

g1ðS; I; PÞ ¼ r 1� Sþ IK

� �� kI � h1P

Sþ d¼ 0; ð11aÞ

g2ðS; I; PÞ ¼ kS� m1P

I2=iþ I þ a� l ¼ 0; ð11bÞ

g3ðS; I; PÞ ¼h2S

Sþ dþ m2I

I2=iþ I þ a� d ¼ 0: ð11cÞ

Straight forward computation show that

S� ¼ dðda�m2I�Þðh2aþm2I� � daÞ ; P� ¼ a

m1ð kdðda�m2I�Þðh2aþm2I� � daÞ � lÞ;

with a ¼ aðI�Þ ¼ ðI� 2=iÞ þ I� þ a:

I⁄ is a positive root of the cubic equation

f ðI�Þ ¼ uI�3 þ vI�2 þwI� þ q ¼ 0 ð12Þ

where

u ¼ ðrm1dh2d� rm1dh22Þ=i;

v ¼ ðrm1dh22K � rm1dh2Kd� rm1d2h2d� dh1kdK þ lh1h2K

� lh1dKÞ=iþ ðrm1dh2d� rm1dh22Þ � ðrm1dh2m2Þ;

w ¼ ðrm1dh22K � rm1dh2Kd� rm1d2h2d� dh1kdK þ lh1h2K

� lh1dKÞ þ aðrm1dh2d� rm1dh22Þ þ ðrm1dh2Km2 þ rm1d2h2m2

� km1Kdh2 þ dh1kKm2 þ lh1Km2Þ;

q ¼ aðrm1dh22K � rm1dh2Kd� rm1d2h2d� dh1kdK þ lh1h2K

� lh1dKÞ:

We note that 0 < I� < K: We f ð0Þ ¼ q < 0 if X < Y , andf ðKÞ ¼ L > 0 if M > N.where

R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187 177

X ¼ h2ðrm1dh2K þ lh1KÞ;Y ¼ dðrm1dh2K þ rm1d2h2 þ dh1kK þ lh1KÞ;

L ¼ M � N;

M ¼ ðK3lh1h2=iÞ þ K2ððrm1dh2d=iÞ þ rm1dh22 þ lh1h2 þ rm1dh2m2

nþdh2km2 þm2h1lÞ þ Kðarm1dh2dþ rm1d2h2m2 þ arm1dh2

2

þalh1h2Þ;g

N ¼ K2ððrm1d2h2d=iÞ þ ðrm1dh22=iÞ þ ðrm1dh2m2=iÞ þ rm1dh2d

nþrm1d2h2dþ dh1kdþ lh1dþm1dh2kÞ þ Kðrm1d2h2dþ arm1dh2

2

þarm1dh2dþ adh1kdþ alh1dþ ðarm1d2h2d=KÞÞo:

Since f ð0Þf ðKÞ < 0 there is a positive root of Eq. (12) lies in ð0;KÞwhen X < Y and M > N are satisfied.

Therefore, the positive equilibrium E4 ¼ ðS�; I�; P�Þ exists underthe following conditions:

m2I� < da < ðh2aþm2I�Þ; ð13aÞ

lh2

ðda�m2I�Þ < ðkdþ lÞ: ð13bÞ

Now, in order to investigate the local behavior of the model sys-tem (4) around each of the equilibrium points, the variational ma-trix V of the point ðS; I; PÞ is computed as

V ¼S @g1

@S þ g1 S @g1@I S @g1

@P

I @g2@S I @g2

@I þ g2 I @g2@P

P @g3@S P @g3

@I P @g3@P þ g3

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3775:

Let Vj; j ¼ 0;1;2;3;4 denotes the variational matrix atEj; j ¼ 0;1;2;3;4 respectively.

For E0; we have

V0 ¼r 0 00 �l 00 0 �d

264

375:

The eigenvalues of V0 are r;�l;�d. There is an unstable mani-fold along S-direction and a stable manifold along IP-direction.Therefore, the equilibrium point E0 is a saddle point.

The variational matrix for E1 is given by

V1 ¼�r �ðr þ kKÞ � h1K

Kþd

0 kK � l 00 0 h2K

Kþd� d

264

375

From the variational matrix V1, it is found that the equilibriumpoint E1 is locally asymptotically stable provided K < l=k;h2 > d and K < ðdd=ðh2 � dÞÞ and E1 is a saddle point ifK > l=k; h2 > d and K > (dd/(h2 � d)).

The variational matrix about the equilibrium point E2 ¼ ð�S;�I;0Þis given by

V2 ¼

� r�SK �ðkþ r

K�S � h1�S

ð�SþdÞ

k�I 0 � m1�I

ðð�I2=iÞþ�IþaÞ

0 0 h1�S

ð�SþdÞ þm2

�Iðð�I2=iÞþ�IþaÞ � d

26664

37775:

The root of the characteristic equation p3ð�qÞ ¼ 0 of the abovevariational matrix about E2 ¼ ð�S;�I;0Þ satisfy the following:

�q1 þ �q2 ¼ �r�SK;

�q1 �q2 ¼ k kþ rK

� ��S�I;

�q3 ¼h1

�Sð�Sþ dÞ

þ m2�I

ðð�I2=iÞ þ�I þ aÞ� d:

The equilibrium point E2 ¼ ð�S;�I; 0Þ is stable or unstable in thepositive direction orthogonal to the SI-plane, i.e. P-direction

depending on whether q3 ¼h1

�Sð�SþdÞ þ

m2�I

ðð�I2=iÞþ�IþaÞ � d is negative or posi-

tive, respectively. The negativity of �q1 and �q2 follows from the pre-vious Section 3.1.

The variational matrix about the equilibrium point E3 ¼ ð��S;0; ��PÞis given by

V3 ¼

��S � rK þ

h1��P

ð��SþdÞ2

� �� kþ r

K

��S � h1��S

ð��SþdÞ

0 k��S� m1��P

a � l 0dh2

��Pð��SþdÞ2

m2��P

a2 0

266664

377775:

The root of the characteristic equation p3ð��qÞ ¼ 0 of the abovevariational matrix about E3 ¼ ð��S;0; ��PÞ satisfy the following:

��q1 þ ��q3 ¼ ��S � rKþ h1

��P

ð��Sþ dÞ2

!;

��q1��q3 ¼

dh1h2��S ��P

ð��Sþ dÞ3;

��q2 ¼ k��S�m1��P

a� l:

The equilibrium point E3 ¼ ð��S;0; ��PÞ is stable or unstable in thepositive direction orthogonal to the SP-plane, i.e. I-directiondepending on whether q2 ¼ k��S� m1

��Pa � l is negative or positive,

respectively. The negativity of ��q1 and ��q3 follows from the previousSection 3.1.

The variational matrix about the equilibrium point E4 = (S⁄, I⁄,P⁄) is given by

V4¼

S� � rKþ

h1P�

ðS�þdÞ2

� �� kþ r

K

S� � h1S�

ðS�þdÞ

kI� m1 I�P�ðð2I�=iÞþ1ÞððI�2=iÞþI�þaÞ2

� m1 I�

ððI�2=iÞþI�þaÞ

dh2P�

ðS�þdÞ2m2P�ða�ðI�2=iÞÞððI�2=iÞþI�þaÞ2

0

266664

377775¼

a11 a12 a13

a21 a22 a23

a31 a32 a33

264

375:

Here a12 < 0; a13 < 0; a21 > 0; a22 > 0; a23 < 0; a31 > 0; a32 > 0 ðifI�2 < aiÞ and a33 = 0.

In the following theorem, we are able to find necessary and suf-ficient conditions for the positive equilibrium point E4 = (S⁄, I⁄, P⁄)to be locally asymptotically stable.

Theorem 2. Suppose that the positive equilibrium point E4 = (S⁄, I⁄,P⁄) of model system (4) exists. The equilibrium point E4 = (S⁄, I⁄, P⁄) islocally asymptotically stable if and only if condition

h2S�

ðS� þ dÞ � d

� �2

<m2½M1M2ðS� þ dÞ2 þ dh2P�M3�M4P�ðð1=iÞ � ða=I� 2ÞÞðS� þ dÞ2

; ð14Þ

where M1¼ða11þa22Þ; M2 ¼ða12a21�a11a22Þ; M3¼ða11a13þa12a23Þ;M4 ¼ ða22a23 þ a13a21Þ hold.

The Proof of this theorem follows from Routh–Hurwitz criterionand hence omitted.

3.3. Hopf bifurcation of model system

In order to investigate the Hopf bifurcation in the model system(4), we follow the technique given by Liu [23]. According to this ap-proach, the simple Hopf bifurcation at r = rc can occur provided

A1ðrcÞ; A3ðrcÞ and wðrcÞ ¼ A1ðrcÞA2ðrcÞ � A3ðrcÞ;

Fig. 4. Basin boundary structure for system (4) computed at different points in Fig. 3(a) for the chaotic attractor at (a) the top-right corner point(l, i) = (4.25, 25); (b) the topmiddle point (l, i) = (3.25, 30); (c) the bottom-left corner point (l, i) = (1.75, 30); all are in the domain �200 6 I; P 6 200. The meanings of the different colors are as follows:Green: color of first attractor, Sky Blue: basin of first attractor, Red: color of second attractor, Maroon: basin of second attractor, Brown: color of third attractor, White: basin ofthird attractor, Dark Blue: color of points that diverges from the screen area, Yellow: color of the chaotic attractor. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

Fig. 2. 2D scan of the model system (4) in (a) ðK; kÞ (b) ðK; dÞ (c) ðK;lÞ parameter spaces. The value of the other parameters are given in Eq. (19) except for a = 18.3.

Fig. 3. 2D scan of the model system (4) in (a) ðl; iÞ (b) ðd; iÞ (c) ðd;lÞ parameter spaces. The value of the other parameters are given in Eq. (19) except for a = 18.3.

178 R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187

are smooth functions of r in an open interval of rc e R such that

(i) A1ðrcÞ > 0; A3ðrcÞ > 0;(ii) w(rc) = A1(rc)A2(rc) � A3(rc) = 0,

(iii) dwðrÞdr

���r¼rc

–0;

where Ai are smooth function of bifurcation parameter r = rc andforms the coefficient of the characteristic polynomial of E4. w isthe principal sub determinant of the Hurwitz matrix of order 3[19].

Now, let d, the total death of predator population is the bifurca-tion parameter. If condition (14) hold together with the followingcondition

dc ¼1

ðS� þ dÞ h2S� � m2½M1M2ðS� þ dÞ2 þ dh2P�M3�M4P�ðð1=iÞ � ða=I�2ÞÞðS� þ dÞ

( )1=224

35: ð15Þ

Then,A1ðdcÞ > 0; A3ðdcÞ > 0 and wðdcÞ ¼ A1ðdcÞA2ðdcÞ � A3ðdcÞ ¼ 0:

Further, it is easy to verify that

Fig. 5. Basin boundary structure for system (4) computed at different points in Fig. 3(b) for the chaotic attractor at (a) the top-right corner point(d, i) = (70, 30); (b) the top-middle point (d, i) = (40, 25); (c) the bottom-left corner point (d, i) = (10, 10). All are in the domain �200 6 I; P 6 200. The meanings of the different colors are same as given infigure caption of Fig. 4.

Fig. 6. Basin boundary structure for system (4) computed at different points in Fig. 3(c) for the chaotic attractor at (a) the top-right corner point (d, l) = (55, 1.75); (b) the top-middle point (d, l) = (35, 1.50); (c) the bottom-left corner point (d, l) = (15, 1.0). All are in the domain �200 6 I; P 6 200 and all these points are in red circles. The meaningsof the different colors are same as given in figure caption of Fig. 4.

R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187 179

dwðdÞdd

����d¼dc

¼ �2M4P�ðh2S� � ðS� þ dÞdcÞða=I�2 � 1=iÞm2ðS� þ dÞ –0:

Theorem 3. Under the conditions (14) and (15), there is a simpleHopf bifurcation of the positive equilibrium point E4 = (S⁄, I⁄, P⁄) of themodel system (4) at some critical value of the parameter d given by(15).

3.4. Nonlinear stability analysis and persistence of the model system

In the following theorem we find sufficient conditions for thepositive equilibrium E4 to be globally asymptotically stable.

Theorem 4. Assuming that the positive equilibrium point E4 = (S⁄,I⁄, P⁄) is locally asymptotically stable. Then it is a globally stable in theinterior of positive octant (i.e., Int R3

þ) provided that

h1g3 < rdðS� þ dÞ=K ; ð16aÞ

m1g2g3I�

iaðI� 2=iþ I� þ aÞþ kg1 < l ; ð16bÞ

h2S�

ðS� þ dÞ þm2I�

ðI� 2=iþ I� þ aÞ< d; ð16cÞ

m1h2ðr þ kKÞkm2dðS� þ dÞ �

h1

ðg1 þ dÞ

� �2

<rK� h1g3

dðS� þ dÞ

� �

d� h2S�

ðS� þ dÞ �m2I�

ðI�2=iþ I� þ aÞ

!: ð16dÞ

Proof. Proof is given in appendix.

In the following, we shall find the conditions for the persistenceof the food chain system (4). h

Theorem 5. Assume that there are no non-trivial periodic solutions inthe boundary plane of system planes of system (4). Then the necessarycondition for the persistence of system (4) is

��q2 ¼ k��S� ðlþm1��P=aÞP 0 ð17aÞ

�q3 ¼h2

�S�Sþ d

þ m2�I

ð�I2=iþ�I þ aÞ� d P 0 ð17bÞ

Fig. 7. Bifurcation diagram of the model system (4), as a function of a is plotted in the range 15 6 a 6 24:5 for the parameters given in Eq. (19) with respect to (a) thesuccessive maxima of S, (b) the successive maxima of I, (c) the successive maxima of P, (d) magnified bifurcation diagram of the successive maxima of S as a function of a isplotted in the range 18:07 6 a 6 18:13.

180 R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187

and the sufficient condition for the persistence of the system is

��q2 ¼ k��S� ðlþm1��P=aÞ > 0 ð17cÞ

�q3 ¼h2

�S�Sþ d

þ m2�I

ð�I2=iþ�I þ aÞ� d > 0 ð17dÞ

Proof. Proof is given in appendix. h

Theorem 6. Suppose that condition (17c) and (17d) hold and thereare a finite number of limit cycles in the SP-plane or in the SI-plane.Then the persistence condition for system (4) takes the formZ T

0g3ð��hðtÞ;0; ��tðtÞÞdt > 0 ð18aÞ

Z T

0g3ð�hðtÞ; �tðtÞ;0Þdt > 0: ð18bÞ

For each limit cycle, ð��hðtÞ; ��tðtÞÞ in SP-plane and each limit cycleð�hðtÞ; �tðtÞÞ in the SI-plane with T is the appropriate period of the limitcycle.

Proof. Proof is given in appendix. h

4. Numerical simulations

In this section, the global dynamical behavior of the model sys-tem (4) is investigated numerically. The objective is to detect the

existence of complex dynamics including chaos in the model sys-tem. Also this chaotic behavior of the model system can be avoidedby implementing a proper harvesting policy which helps us in resto-ration of the damaged ecosystem. There are many ways to detect thechaos in dynamical system. For investigating the deterministicbehavior of the model system we have used the 2D scan, bifurcationanalysis, largest Lyapunov exponent and basin boundary computa-tions. The parameter values are chosen on the basis of biologicalprinciples such that the system is biologically feasible. The rangesof values of the parameters are chosen on the basis of the values re-ported in Jorgensen [16]. It is observed that for the following biolog-ically feasible set of parameter values, model system (4) havedifferent type of attractors with variation of parameter a, the halfsaturation constant in the absence of inhibitory effect, as shown inFig. 1. The model was numerically integrated to get the time seriesfor the infected fish population, I, of the model system (4) ata ¼ 16 and is presented in Fig. 1(e). The other parameters are fixed at

r ¼ 22;K ¼ 75; k ¼ 0:4; h1 ¼ 1; d ¼ 60;m1 ¼ 15:5; i ¼ 40;l ¼ 3:4; h2 ¼ 4; m2 ¼ 8:1; d ¼ 3:6: ð19Þ

Since every nonlinear system has finite amount of transients,the data points representing transient behavior were discarded.3D phase portraits were drawn using this data to obtain the geom-etry of the attractors.

We have performed 2D parameter scan to identify the parame-ter regimes in which chaotic dynamics exists. The basis of 2D scanis the belief that the changes in physical/eco-epidemiologicalconditions may bring corresponding changes in at most twoparameters at a time. The changes in the nature of dynamics are

Fig. 8. Bifurcation diagram of the model system (4), as a function of i is plotted in the range 23:0 6 i 6 50:0 for the parameters given in Eq. (19) with a = 18.3 with respect to(a) the successive maxima of S, (b) the successive maxima of I, (c) the successive maxima of P, (d) magnified bifurcation diagram for the successive maxima of S as a functionof i is plotted in the range 37:5 6 i 6 39:0.

R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187 181

monitored. The parameters used for 2D scans are K; k; d; i and l.We have presented the 2D scan diagrams in various parameterspaces in Figs. 2 and 3. From Fig. 2, we find that chaotic dynamicsis confined to a dense region in the parameter spaces. This is cer-tainly not a manifestation of short term recurrent chaos (STRC).The robust chaos is localized to the dense region of the parameterspaces ðK; kÞ, ðK; dÞ and ðK; lÞ with step size of K ¼ 5; k ¼ 0:01;d ¼ 5 and l ¼ 0:25. This represents that the parameter spaces cre-ated by the carrying capacity K with the other parametersk; d; and l play an important role to depict the chaotic dynamicsof the model system (4). We also present 2D scan diagrams in otherparameter spaces like ðl; iÞ, ðd; iÞ and ðd;lÞ in Fig. 3 with the mid-dle point ðl; iÞ ¼ ð3:25;30Þ, ðd; iÞ ¼ ð40;25Þ and ðd;lÞ ¼ ð35;1:50Þat which basin boundary calculation are performed together withtop right and bottom left corner points.

The computation of basin boundaries of coexisting attractors isuseful for our discussion. For a dynamical behavior to be of any prac-tical value it is essential that it should exist in a wide parameter re-gime and corresponding natural measure in 2D parameter scanshould be nonzero. In addition, it must fulfill the requirement thatit should possess a phase space of initial condition whose naturalmeasure is nonzero. When these two conditions are met for a partic-ular dynamical behavior, then the same is understood to be a robust

one and is considered to have some significance. In this case, stabil-ity demands that the basin boundary of the coexisting attractorsshould be smooth (no-fractal). The basin boundary calculations formodel system (4) are represented in Figs. 4–6. In these figures wehave presented the SP-view (�200 6 S 6 200; �200 6 P 6 200Þ ofthe basin boundary structure of chaotic attractor (shown in yellowcolor). The basin boundary calculations are performed using the ba-sins and attractors structure routine developed by Maryland Chaosgroup. We have used the ‘‘Dynamics’’ software package of Nusseand Yorke [28], for all the basin boundary calculations. It is clearfrom these figures that basin boundaries of the chaotic attractorare fractal. This indicates presence of dynamical complexities inthe eco-epidemiological system (4). It is also seen that basin ofattraction of different attractors are intermixed. The encroachmentinto the basin of chaotic attractor by basin of attractor at infinity(shown in green color) can be observed in Figs. 4–6. It appears be-tween the first attractor (shown in green color) and its basin (shownin sky blue color). The interesting feature in the system (4) is that theriddled basin with fractal boundary lies in the basin of repeller whichhas many rectangular and square holes created by chaotic attractor.This complicated basin boundary structure suggests that the systemdynamics may have loss of even qualitative predictability in the caseof external disturbances. These figures also suggest that a change in

Fig. 9. Bifurcation diagram of the model system (4), as a function of k is plotted in the range 0:05 6 k 6 0:85 for the parameters given in Eq. (19) with a = 18.3 with respect to(a) the successive maxima of S, (b) the successive maxima of I, (c) the successive maxima of P.

182 R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187

the parameters l; i and d introduce a transition in the dynamicalbehavior of the model system and suggest that the system dynamicsis also sensitive to changes in initial conditions. This implies that sto-chastic external influences dictate the dynamical behavior of thishybrid system. Since chaotic dynamics exists in discrete mannerand the dynamical behavior of this system under the influence ofexogenous factors may be unpredictable as intertwined basinboundaries are common. Distribution of points in the parameterspace (Fig. 3) suggests that system (4) displays STRC. It is character-ized by chaotic bursts repeated at unpredictable intervals.

The bifurcation analysis of the model system (4) unravels richand complex dynamics are observed, involving various sequencesof period-doubling bifurcation leading to chaotic dynamics or se-quences of period-halving bifurcation leading to limit cycles[36,42]. For bifurcation diagram of model system (4) presented inFigs. 7(a)–(c), the successive maxima of S; I; P in the ranges12 6 S 6 75, 0:05 6 I 6 35 and 15 6 P 6 45 respectively as a func-tion of a; the half saturation constant in the range 15 6 a 6 24:5and the other parameters are given in the Eq. (19). Fig. 7(d) repre-sents the magnified bifurcation diagram of the successive maximaof S in the range 25 6 S 6 75 as a function of a is plotted in therange 18:07 6 a 6 18:13: Similarly, we can also obtain the magni-fied bifurcation diagram of the successive maxima of I and P as afunction of parameter a. The magnified bifurcation diagram showsthat the system possesses a rich variety of dynamical behavior.

Closed curve in this diagram correspond to invariant KAM tori inthe phase space. Later on, theses curves break and give rise to cha-otic dynamics. The chaotic behavior of the system is not continuingfurther, as the unstable period-3 orbits which originate at the timeof saddle-node bifurcation do not allow it to move further.

The other bifurcation diagrams are generated for the successivemaxima of population densities of S; I; P in the ranges 15 6 S 6 75;0:05 6 I 6 35 and 22 6 P 6 45 respectively as a function of predator’simmunity parameter from the infection, i in the range 23 6 i 6 50 andwith a ¼ 18:3 (see Fig. 8(a)–(c)). Fig. 8(d) represents the magnifiedbifurcation diagram of the successive maxima of S in the range25 6 S 6 75 and as a function of i in the range 37:5 6 i 6 39. Fig. 9 rep-resents the bifurcation diagrams generated for the successive maximaof the population densities S; I; P in the ranges 25 6 S 6 75,0:02 6 I 6 30 and 0:02 6 P 6 70 respectively as a function of the rateof transmission, k in the range 0:05 6 k 6 0:85 and value of otherparameter are given in Eq. (19) with a ¼ 18:3.

The increase in size of a chaotic attractor as the system param-eter is varied is considered to be the hallmark of the crisis (suddendestruction of a chaotic attractor) route to chaotic dynamics[37,38]. The crisis occurs precisely at the point where the unstableperiod-3 orbit created at the original saddle – node bifurcationintersects with the narrow chaotic region. The other importantchanges in the chaotic set include interior crisis in which a chaoticattractor undergoes a sudden increase in the size [15] along with

Table 2Simulation experiments of model system (5) with the fixed parameter values r = 22, K = 75, k ¼ 0:4; h1 = 1, a ¼ 18:3; d = 60, m1 = 15.5, i = 40, l = 3.4, h2 = 4, m2 = 8.1, d = 3.6 andinitial condition (S0, I0, P0) = [10,10, 10] and SS = Step Size.

Parameters kept constant Parameter varied Ranges in which parameter was varied Dynamical behavior

K; k; a; i; d;l r (SS = 0.05) 5.0–7.70 Extinction5 6 r 6 35 7.75–8.45 Chaos

8.50–35.0 Stable limit cycler; k; a; i; d;l K (SS = 5) 25–30 Extinction

25 6 K 6 500 35–60 Stable focus65–170 Stable limit cycle175–500 Chaos

r;K; a; i;d;l k (SS = 0.01) 0.01–0.04 Extinction0.01 6 k 6 2.0 0.05–0.07 Stable focus

0.08–0.76 Stable limit cycle0.77–0.84 Chaos0.85–2.0 Extinction

r;K; k; i; d;l a (SS = 0.1) 5.0–16.4 Chaos5 6 a 6 25 16.5–22.7 Stable limit cycle

22.8–25.0 Stable focusr;K; k; a; d;l i (SS = 1) 10–22 Extinction

10 6 i 6 75 23–26 Chaos27 Stable limit cycle28–30 Chaos31–75 Stable limit cycle

r;K; k; a; i;l d (SS = 0.1) 10.0 Stable limit cycle10 6 i 6 75 10.1 Chaos

10.2 Stable limit cycle10.3–10.6 Chaos10.7–10.8 Stable limit cycle10.9 Chaos11.0–11.1 Stable limit cycle11.2–11.4 Chaos11.5–100.0 Stable limit cycle

r;K; k; a; i;d; l (SS = 0.1) 1.0–1.2 Stable limit cycle1.0 6 l 6 5.0 1.3–1.6 Chaos

1.7–5 Stable limit cycle

R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187 183

the appearance or sudden enlargement of a fractal structure in thebasin boundary. From the bifurcation diagrams (Figs. 7–9), it wasobserved that the sudden increase in the size of chaotic attractorfor susceptible and infected population are in opposite directionsand for the predator population it is in both sides of the attractor.It depends on the nature of interaction by which predator predatesthe prey (Holling type II or IV).

Simulation experiments were performed to determine the re-gions in the parameter spaces, which support different dynamicalbehavior in the model system (4). The computed results are givenin Table 2, which are the main results of the complex dynamicalbehavior of the model system (4). From Table 2, it is found thatchaos was observed in the ranges 7:75 6 r 6 8:45; 175 6 K 6 500;0:77 6 k 6 0:84; 5 6 a 6 16:4; 23 6 i 6 26; 28 6 i 6 30; 1:3 6 l 61:6; 10:3 6 d 6 10:6; 11:2 6 d 6 11:4 and at the discrete pointsd ¼ 10:1; d ¼ 10:9 and other parameters are fixed at the limit cyclevalue i.e., h1 ¼ 1; m1 ¼ 15:5; h2 ¼ 4; m2 ¼ 8:1. Extinction of preda-tor population is also observed for some of the parameter values innarrow ranges i.e., 5:0 6 r 6 7:70; 25 6 K 6 30; 0:01 6 k 6 0:04;

0:85 6 k 6 2:0; 10 6 i 6 22 and the regular dynamics (stable limitcycle and stable focus) are obtained for rest part of these parame-ters. From Table 2, we learn that chaos is detected in the larger rangeof carrying capacity K, in the range 175 6 K 6 500. The parameterwhich can be controlled easily is k, the transmission rate of theinfection from the infected fishes to susceptible once. There is aneed to identify and remove infected fishes by mechanical meansout of the sea at regular intervals. The periodicity of the removalis decided by the type of infection, its rate of progression in individ-ual fishes from acute to chronic phases. Fig. 10 presents the largestLyapunov exponent which confirm the chaotic dynamics of themodel system for the parameter values a and l in the ranges5 6 a 6 16:4 and 1:3 6 l 6 1:6 respectively as reported in Table 2.

5. Application to Salton Sea

We now apply model system (4) to the case study of Salton Seawetland of California, USA. California’s crown jewel of avian diver-sity is in danger. The deterioration in quality of the agriculturalrun-off which sustains this sea has reached a level which is threat-ening the existence of this ecosystem. More than 90 per cent of thewetlands of California have been lost [34]. The importance of thesea as a habitat for inland wetland species has increased withthe decline of California’s wetlands. This is why some scientists callthe Salton Sea ‘‘California’s crown jewel of avian biodiversity’’. Welist the major challenges the Salton Sea Authority faces in restoringthe sea to good health:

(i) Rising salt level in the lake,(ii) Run-off from farms surrounding the Salton Sea causes seri-

ous eutrophication problems. The associated drop in oxygenlevel is endangering the life of the fish in the sea.

(iii) Outbreaks of diseases; e.g., avian botulism and avian cholerakill thousands of birds each year.

The strategy of recovery for the Salton Sea ecosystem is based onour realization of the fact that chaotic dynamics is generally associ-ated with excursions to extinction–sized densities [1,30]. The exoge-nous stochastic factors (e.g., outbreak of an epidemic, abrupt climatechanges) influence the system dynamics most when it makes suchexcursions. In certain cases it may lead to extinction of one of theconstituent population. 2D scans (Figs. 2(c) and 3(c)) show that l, to-tal death rate of infected Tilapia should be brought down in order toavoid chaotic dynamics. This would happen when the quality of agri-cultural run-off to the Salton Sea is regulated so that the dissolvedoxygen level in the sea water improves.

Fig. 11. (a) This figure shows the plot between time (years) vs. number of dead and sick Pelicans due to botulism at the Salton sea during the period 1994–2001 and 1996–2001 respectively, and (b) represents the how the fluctuation of dead and sick Pelicans is running during the time 1994–2001 and 1996–2001 respectively. Data is derivedfrom Rocke et al. (2004).

Fig. 10. The Lyapunov exponent bifurcation diagram of system (4) as a function of (a) a in the range 5 6 a 6 25 (b) l in the range 1 6 l 6 5, keeping the rest of parameters asgiven in Eq. (19).

184 R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187

Fig. 12. Percentage of dead Tilapia collected in 1996 that was positive for type C botulinum toxin. (a) presents the different pockets of Salton Sea in which different places arefound with different percentage of dead Thilapia Fish, (b) represents the fluctuation of dead Tilapia fish (in %) with different places.

R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187 185

Fig. 11 represents the Pelicans affected by botulism at the Sal-ton Sea, during the period 1994–2001. We have also presentedthe Tilapia Sampling 1996 in Fig. 12. The figure shows differentplaces around the Salton Sea where dead Tilapia fish was foundin the year 1996.

In nonlinear dynamical systems, several attractors co-exist atthe same set of parameter values. Basin boundary diagrams pro-vide us an idea about possibilities of dynamical transitions causedby exogenous external influences (e.g., outbreak of an epidemicwhich occurred in year 1996 shown in Fig. 11(a) as sudden risein the number of Pelicans) which act directly on the populationdensities. Fig. 11(a) represents the number of dead and sick Pelicanto botulism of Salton Sea during the period 1994–2001. We havecomputed basin boundary structures for chosen set of parametervalues. Fig. 12 represents the percentage of dead Tilapia collectedin 1996 that was positive for type C botulinum toxin. Fig. 11(b).represents the fluctuation in dead and sick Pelicans due to botu-lism at the Salton Sea, running during the period 1994–2001 and1996–2001 respectively. No data is available for sick Pelicans dur-ing the period 1994–1995. These Figs. 11 and 12 are based on thereal data collected from Rocke et al. [31]. The restoration of the Sal-ton Lake provides a perspective for conservation and managementstrategy, since the recovery phase of large, deep, hard water andbiological quality of lake [35] depends on the contact opportunitiesbetween susceptible and infected population. The model predictsnon-linear oscillations in all the constituent species of the system(cf. Fig. 1(d)). The oscillations in number of sick Pelicans (cf.Fig. 11(b)) are driven by those in infected Tilapia which is a staplediet for Pelican birds. If we compare the Figs. 1(d) and Fig. 3, thenumerical results (generated for model set of parameter values)to the Figs. 11(b) and 12(a) drawn from the real data set given in[31,32], then it shows the fluctuation/oscillatory behaviour indynamics of model system matches during certain period of times.Thus, our model explains the data from the Salton Sea.

6. Discussion and conclusion

If one examines the simulation results carefully, one finds thatthere are three key parameters which control the dynamics ofthe designed model system: (i) transmission rate of infection, (k),(ii) rate of death of infected Tilapia due to algal boom, (l2) and(iii) the immunity of the Pelicons from the infection (i). The secondparameter can be understood in terms of mortality of the infectedTilapia because of another cause which is ecological in origin:eutrophication. It essentially causes scarcity of dissolved oxygenin the sea water without which fishes cannot survive. The rate ofmortality would also depend on the type of infection and opportu-

nity for progression in the individuals of the fishes under consider-ation. It could be Tilapia or any other type. Our choice of Tilapia isdecided solely on the basis of availability of the relevant data. Thethird parameter (i), i.e., the immunity of the birds from the infec-tion which infected fishes carry, is not a constant. It will evolveto higher values with time. As it can be learnt from the 2D scans(Figs. 3(a) and (b)), it favors chaotic dynamics and from the Table 2we observed that chaotic dynamics is obtained in the ranges23 6 i 6 26 and 28 6 i 6 30. For i P 31; model system exhibits sta-ble limit cycle behavior and for lower value of i 2 ½10;20�, the sys-tem favors extinction phenomena. Therefore, we need tomanipulate the other relevant parameters; e.g., l, total death rateof infected prey, and d, which measures the extent to which theenvironment provides protection to susceptible prey. The valueof parameter d depends on the agility of the fishes and the avail-ability of refuges to avoid sightings from the birds. The conserva-tion workers can control l more easily. The parameter d cannotbe manipulated. Moreover, chaotic dynamics exists for broad rangeof parameter d and K (see Fig. 2(b)). Fig. 3 suggests that it should beexecuted at the time when the infected population density is onthe rise in order to extract maximum benefit from these efforts.This is the only way to reduce the transmission rate.

Bifurcation diagrams give an idea about Hamiltonian and Dissi-pative dynamical mechanisms working together. A close scrutinyof bifurcation diagrams (ones obtained after magnification of cer-tain selected sets of the parameter range mentioned in the figurecaptions) show that Hamiltonian and Dissipative dynamics are atwork one after the other.

The restoration and recovery efforts for the Salton Sea shouldfocus on bringing the opportunities of contact between the suscep-tible and infected fishes to a minimum. This can be achieved byidentifying and removing the infected ones at regular interval oftime. The other ecologically feasible intervention involves regulat-ing the quality of agricultural run-off to the sea.

Appendix A

Proof of Theorem 4. Consider the following positive definitefunction about the equilibrium point E⁄:

VðtÞ ¼ S� S� � S� lnSS�

� �� �þ k1

2ðI � I�Þ2 þ k2

2ðP � P�Þ2 :

where k1 and k2 are positive constants to be chosen suitably lateron.

Now differentiating V with respect to time t along the solutionsof the model system (4), a little algebraic manipulations yields

186 R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187

dVdt¼� r

K� h1PðSþdÞðS� þdÞ

� �ðS� S�Þ2

� k1 l� kSþ m1Pða� II�=iÞðI2=iþ IþaÞðI�2=iþ I� þaÞ

!ðI� I�Þ2

� k2 d� h2S�

ðS� þdÞ�m2I�

ðI�2=iþ I� þaÞ

!ðP�P�Þ2

þðS� S�ÞðI� I�Þ k1kI� � ðkþ r=KÞð Þ

þ ðI� I�ÞðP�P�Þ k2m2Pða� II�=iÞðI2=iþ IþaÞðI�2=iþ I� þaÞ

� k1m1I�

ðI�2=iþ I� þaÞ

!

þðS� S�ÞðP�P�Þ k2h2dPðSþdÞðS� þdÞ�

h1

ðSþdÞ

� �:

The above equation can be written as sum of the quadratics

dVdt¼ �1

2a11ðS� S�Þ2 þ a12ðS� S�ÞðI � I�Þ � 1

2a22ðP � P�Þ2

� 12

a22ðI � I�Þ2 þ a23ðI � I�ÞðP � P�Þ � 12

a33ðP � P�Þ2

� 12

a11ðS� S�Þ2 þ a13ðS� S�ÞðP � P�Þ � 12

a33ðP � P�Þ2;

where

a11 ¼rK� h1PðSþ dÞðS� þ dÞ

� �; a12 ¼ ðk1kI� � ðkþ r=KÞÞ;

a13 ¼k2h2dP

ðSþ dÞðS� þ dÞ �h1

ðSþ dÞ

� �;

a22 ¼ k1ðl� kSþ m1Pða� II�=iÞðI2=iþ I þ aÞðI� 2=iþ I� þ aÞ

Þ;

a33 ¼ k2 d� h2S�

ðS� þ dÞ �m2I�

ðI� 2=iþ I� þ aÞ

!;

a23 ¼k2m2Pða� II�=iÞ

ðI2=iþ I þ aÞðI� 2=iþ I� þ aÞ� k1m1I�

ðI� 2=iþ I� þ aÞ

!:

Sufficient conditions for dVdt to be negative definite are that the

following inequalities hold:

a11 > 0; a22 > 0; a33 > 0; ð20aÞ

a212 < a11a22; ð20bÞ

a223 < a22a33; ð20cÞ

a213 < a11a33: ð20dÞ

It can be seen that under condition (16a), a11 > 0. Under condi-tions (16b), a22 > 0; and under condition (16c), a33 > 0 and undercondition (16d) the condition (20d) hold. If we choosek1 ¼ ðrþkKÞ

kKI� and k2 ¼ k1m1 I�

m2g3, then it can be checked that conditions

(20b) and (20c) are automatically satisfied. h

Proof of Theorem 5. Since the boundedness is proved and also ��q2

and �q3 is the eigenvalue which gives the stability in the positivedirection orthogonal to the SP-plane and SI-plane. Thus, if thereare no non-trivial periodic solution in the SP-plane and if ��q2 < 0;then there is an orbit in the positive cone which approachsE3 ¼ ð��S;0; ��PÞ. Therefore, (17a) is one of the necessary conditionsfor the persistence. Similarly, with respect to E3 ¼ ð�S;�I;0Þ in theSI-plane the other necessary condition (17b) holds.

Now for the sufficient condition of persistence of the modelsystem (4), we shall apply the abstract theorem of Freedman andWaltman [13]. According to the growth functions g1, g2 and g3 ofsystem (4) the following hypothesizes are satisfied

(A1)@g1@I < 0; @g1

@P < 0; @g2@S > 0; @g3

@S > 0; @g3@I > 0;

g2ð0; I; PÞ < 0; g3ð0;0; PÞ < 0; g3ð0;0; PÞ > 0:

(A2) The susceptible species S grows to carrying capacity in theabsence of predation, g1ð0;0;0Þ > 0 and g1ðK;0;0Þ ¼ 0.While, due to the transmission rate between susceptibleand infected species, we have @g1

@S ðS;0;0Þ < 0: However thepredation population dies in the absence of the susceptibleand infected prey i.e., g3ð0;0;0Þ < 0.

(A3) There is no equilibrium point in IP-plane.(A4) In the absence of susceptible fish, the predator population

cannot survive on infected fish because in the absence sus-ceptible fish, infected fish decay exponential. This is alwaystrue under the Kolmogorov condition. Therefore E2 ¼ ð�S;�I;0Þand E3 ¼ ð��S;0; ��PÞ always exist in the SI-plane and SP-planerespectively, such that g1ð�S;�I;0Þ ¼ g2ð�S;�I;0Þ ¼ g1ð��S;0; ��PÞ ¼g3ð��S;0; ��PÞ ¼ 0:

Hence, due to the above hypothesis there is at most one planerequilibrium point in each positive coordinate plane. Therefore,according to abstract theorem of Freedman and Waltman [13],conditions (17c) and (17d) are sufficient conditions for persistenceof system (4). h

Proof of Theorem 6. Let SðtÞ ¼ ��hðtÞ; IðtÞ ¼ 0 and PðtÞ ¼ ��tðtÞ a limitcycle in the SP-plane, then the variational matrix V aboutð��hðtÞ;0; ��tðtÞÞ take the form

Vð��hðtÞ;0;��tðtÞÞ¼

��hðtÞ@g1@S þg1ð��hðtÞ;0;��tðtÞÞ ��hðtÞ@g1

@I��hðtÞ@g1

@P

0 g2ð��hðtÞ;0;��tðtÞÞ 0��tðtÞ@g3

@S��tðtÞ@g3

@I 0

0BB@

1CCA;ð21Þ

where all the partial derivative and gj ; j ¼ 1; 2; 3 in (21) are com-puted at the limit cycle ð��hðtÞ;0; ��tðtÞÞ. Consider a solution of themodel system (4) with positive initial condition ðt;a1;a2;a3Þ suffi-ciently close to the limit cycle.

From the variational matrix (21), @I@a2

is a solution of the system

dIdt¼ ½g2ð��hðtÞ;0; ��tðtÞÞ�I

with I(0) = 1.Thus,

@I@a2ðt;a1;a2;a3Þ ¼ exp

Z t

0g2ð��hðtÞ;0; ��tðtÞÞds

� �: ð22Þ

Then using Taylor expansion, we have

Iðt;a1;a2;a3Þ ¼ Iðt;a1;0;a3Þ þ@I@a2

a2 þ � � �

Substituting in Eq. (22) yields

Iðt;a1;a2;a3Þ � Iðt;a1;0;a3Þ ffi a2 expZ t

0g2ð��hðtÞ;0; ��tðtÞÞds

� �:

Then I increase or decrease according toR T

0 g2ð��hðtÞ;0; ��tðtÞÞdt ispositive or negative, respectively. Since E3 and these limit cyclesare the only possible limit in the SP-plane of trajectories with posi-tive initial condition. Hence the trajectories go away from the SP-plane if conditions (17c) and (18a) hold. Similar arguments applyfor (17d) with (18b). This completes the proof. h

R.K. Upadhyay et al. / Mathematical Biosciences 242 (2013) 172–187 187

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