PLANT RESPONSES TO DISEASE AND HERBIVORE ATTACK: A MATHEMATICAL MODEL

This paper presents a model for plants subject to a disease, harvested by (insect) herbivores and it also includes natural enemies of the latter. Basic results on boundedness, feasibility of equilibria, uniform persistence and local and global stability issues are investigated.


Introduction
In nature, plants interact in complex ways with herbivores and pathogens.Presently, 50% of 6 million insects species are herbivorous [17].Plant pathogenic microbes are not estimated still they create major threats to plants [19].Due to multiple attack, plants have developed sophisticated defence mechanism that allow them to identify herbivores or pathogens [13].Two types of defence mechanism are employed by plant namely physical and chemical.In case of chemical defence, plant release volatile organic compounds (VOCs) that attract natural enemies E-mail address: mukherjee1961@gmail.com Received December 30, 2017 of herbivores to reduce enemy pressure [6,16].In fact, VOCs mainly attract predatory mites and parasitic wasps.For example, lima bean and apple plants which release volatile that attract predatory mites when damaged by spider mites [20].Several plant species such as cucumber, corn, cotton etc. release volatile when they are attacked by herbivores.This herbivore-induced plant volatile (HIPV) can control pest and reduce the use of artificial pesticides.HIPVs are the lipophilic liquids with high vapor pressures which are released from the different parts of the plant body namely leaves, flowers fruits etc. [7].Thus the induced plant can protect forestry and agriculture.Though, VOCs can attract predatory arthropods and / or repel herbivores and thus promote plant fitness [1,5,21].Volatile of any kind confer protection for damaged plants by attracting natural enemies of herbivores [15,16].Studies either on plant-herbivore or plantpathogen interaction has received much attention to the researchers.But the role of VOCs in mediating tritrophic interactions, when plants are attacked by herbivore and pathogen is still unknown.Recently Liu et al. [11] investigated the model involving plants, herbivores and natural enemies of herbivores in the form of tritrophic interactions without considering pathogenic effect on the plant population.They found that increase in attraction of strength of plant-induced volatile to the natural enemy leads to high fluctuation amplitude of plant biomass and herbivore population.It was observed that when the attack strength of natural enemies reaches a certain level, fluctuation amplitude of plant biomass and herbivore population decrease and plant biomass approach to its environmental carrying capacity.Fergola and Wang [8] improved the model of Liu et al. [11] and considered the effect of time delay.They established that for Volterra type interaction, the threshold value for persistence of herbivore and carnivore populations is not affected by the chemical attractions.It was remarked that presence of carnivores may decrease the density of herbivores and increase the density of plants.The model exhibits the fold bifurcation when the predation process follows Leslie type.
In view above discussion, we consider a system where pathogens co-occur with herbivory.The novelty of our work is to demonstrate the situation when plants are attacked by herbivorous insects and pathogens.Previous studies mainly focussed plant-insect and plant-pathogen interactions largely independent from one another, although plant defence against both is regulated by the same general mechanism.
One of the main findings of this work is the condition under which the positive (endemic) equilibrium is globally asymptotically stable.For this condition, the disease become endemic.
To control disease such condition should not be accepted.On the other hand, one may remove infected plants to prevent disease.Local stability of boundary equilibria are derived.If these conditions are satisfied then one of the subpopulation faces extinction.In that case, either infected plants or herbivore population can be removed.To protect plant fitness, immigration rate of natural enemies of herbivores and pathogens play a major role.For a four species system, one critical condition demands that all boundary equilibria with one missing species can be invaded by the missing species.These conditions are obtained in system parameters.This paper is structured as follows.In Section 2, we present our model.Positivity and boundedness of solutions of system are given in Section 3. Dynamical behavior of the system are investigated in Section 4. Uniform persistence criterion is described in Section 5. We discuss global stability of the positive equilibrium point in Section 6.A brief discussion follows in Section 7.

Model
Let S(t), I(t) be the number of susceptible and infected plant respectively.Y (t) and Z(t) be the population sizes of herbivores and their natural enemies respectively.The model is described by : with initial conditions given by S(0) = S 0 > 0, I(0 Here r is the intrinsic growth rate of plants.k is the environmental carrying capacity.β is the disease transmission rate.p 1 and p 2 are the predation rates for plant-herbivore and herbivorenatural enemies respectively.c 1 and c 2 are the corresponding conversion rates.d 1 is the death rate of herbivores, and µ is the death rate of infected plant.The term ω is the immigration rate of natural enemies of herbivores due to the attraction of defensive chemical from plants. Biological justification of the above model can be found in [3].For example, lima bean plants (Phaseolus lunatus L. cv Sieva ) emit volatile when attacked by spider mites (Tetranychus urticae) and attract predatory mite (Phytoseiulus persimilis), a specialized natural enemy of spider mites.Pod blight is caused to the lima bean plants by the fungus Diaporthe phaselorum.
All the model parameters are assumed to be positive.

Prelimineries
In this section, we shall first show positivity and boundedness of solutions of system (2.1).
For system (2.1) to be biologically meaningful and well posed, it is necessary to prove that all solutions of system with positive initial data will remain positive for all times t > 0. This will be established by the following lemma.
Proof.The positivity of S(t), I(t) and Y (t) can be verified by the equations The positivity of Z(t) can be easily deduced from the fourth equation of system (2.1).We observe that then S(t) > 0 for all t > 0. The same argument is valid for component

I(t),Y (t) and Z(t). Hence the interior of R 4
+ is an invariant set of system (2.1).In the theoretical eco-epidemiology, the boundedness of the system ensures that the system is biologically valid and well behaved.Biological validity of the model is shown by the following lemma.

Dynamical behaviour
Evidently, system (2.1) has the following boundary equilibrium points : ) and E 134 = ( S, 0, Ỹ , Z) where S is the positive root of the equation Clearly, E 0 and E 14 always exist.
The local stability of these equilibria is determined by the eigenvalues of the Jacobian matrix of (2.1), (iv)E 134 is locally stable if β S < µ.
Proof.It follows immediately by linearizing around the equilibria.
Next we interested about the existence of the interior equilibrium point of system (2.1) which is given by E * = (S * , I * ,Y * , Z * ) where whose characteristic equation is where From the Routh-Huwitz criteria, all the real parts of roots for (4.1) are negative if and only if If the inequality (4.2) holds then E * is locally asymptotically stable.

Uniform persistence
Biologically, uniform persistence of the system ensures the long term survival of all populations, none of them facing extinction.To establish uniform persistence of system (2.1) we use Butler-McGehee lemma [10].Now we state coexistence condition for all the populations.
(iv) the equilibrium points E 124 and E 134 are globally stable with respect to R + SIZ and R + SY Z respectively.
Proof.Suppose that x is a point in the positive octant and o(x) is the orbit through x and Ω is the omega limit set of the orbit through x.Note that Ω(x) is bounded.We show that E 0 / ∈ Ω(x).If E 0 ∈ Ω(x), then by Butler-McGehee lemma [10], there exists a point p in Ω(x) ∩W s (E 0 ) where W s (E 0 ) denotes the strong stable manifold of E 0 .Since o(p) lies in Ω(x) and W s (E 0 ) is the I − Y − Z plane and hence unbounded orbits lies in Ω(x) is unbounded, which is a contradiction.
Next we show that E 14 / ∈ Ω(x).The condition k > max{ µ β , d 1 c 1 p 1 }, implies that E 14 is a saddle point.W s (E 14 ) is the S − Z plane.Thus unbounded orbits lies in Ω(x) once more a contradiction.The condition is the S − I − Z plane.So one easily show that unbounded orbits lies in Ω(x) once more a contradiction.Lastly, we can show that E 134 / ∈ Ω(x) as β S > µ.Thus , Ω(x) does not intersect any of the coordinate planes and hence system (2.1) is persistent.Since (2.1) is bounded, by main theorem in [4], this implies that the system is uniformly persistent.

Global stability of positive equilibrium point
If the inequality (4.2) holds then the positive equilibrium point E * is locally asymptotically stable under certain restrictions.So natural question arises under what additional conditions it becomes globally asymptotically stable.To derive global stability condition it is sometimes difficult to find out a Lyapunov function.There is an alternative approach to show global stability due to Li and Muldowney [12].Now we use a high-dimensional Bendixson's criterion of Li and Muldowney [12], which we briefly state next.
Let D ⊂ R n be an open set and F ∈ C 1 .Consider a system of differential equations dX dt = F(X).(6.1) According to the theory developed in [12], it is sufficient to show that the second compound equation with respect to a solution X(t, X 0 ) of system (6.1) is equi-uniformly asymptotically stable, namely, for each X 0 ∈ D, system (6.2) is uniformly asymptotically stable, and the exponential decay rate is uniform for X 0 in each compact subset of D, where D ⊂ R n is an open connected set.Here ∂ F/∂ X [2] is the second additive compound matrix of the Jacobian matrix ∂ F [2] /∂ X.
It is an  matrix, and thus (6.2) is a linear system of dimension Fiedler [9] and Muldowney [14]).For a general 4 × 4 matrix its second compound matrix P [2] is 3) The equi-uniform asymptotic stability of (6.2) implies the exponential decay of the surface area of any compact two-dimensional surface D. If D is simply connected, this prevents the occurrence of any invariant simple closed rectifiable curve in D, including periodic orbits.The following result is proved in Li and Muldowney [12].
Proposition 6.1.Let D ⊂ R n be a simply connected region.Assume that the family of linear systems (6.2) is equi-uniformly asymptotically stable.Then (i) D contains no simple closed invariant curves including periodic orbits, homoclinic orbits, heteroclinic cycles; (ii) each semi-orbit in D converges to a single equilibrium.
In particular, if D is positively invariant and contains an unique equilibrium X, then X is globally asymptotically stable in D.
One can show uniform asymptotic stability of system (6.2) by constructing a Lyapunov function.
For example, (6.2) is equi-uniformly asymptotically stable if there exists a positive definite function V (U), such that dV (U)/dt| (6.2) is negative definite, and V and dV (U)/dt| (6.2) are both independent of X 0 .
We now require the following assumptions to prove the global stability of positive equilibrium point of system (2.1).
(A 1 ) There exist positive numbers α, θ , η, ρ, and σ such that Assumption (A 2 ) implies that system (2.1) is uniformly persistent and hence there exists a time We again denote X = (S, I,Y, Z) T and F(X) = (S{r(1 We have ∂ F ∂ X = J(S, I,Y, Z) and by (6.3) and we assume that where The second compound system where X(t) = (S(t), I(t),Y (t), Z(t)) T is arbitrary solution of system (2.1) with The direct calculations lead to the following inequalities : where d + /dt denotes the right-hand derivative and θ + c 55 , c 62 σ η + c 63 σ + c 66 }.Thus, under assumptions (A 1 ) and (A 2 ) and by the boundedness of solutions of system (2.1), there exists a positive constant ξ such that ψ ≤ −ξ < 0 and thus This proves the equi-uniform asymptotic stability of the second compound system (6.5), and hence the positive equilibrium point E * of system (2.1) is globally stable due to Proposition 6.1.
From above analysis, we now state the sufficient conditions for the global asymptotic stability of the positive equilibrium.Theorem 6.1 If the assumptions (A 1 ) and (A 2 ) are satisfied then system (2.1) has no non-trivial periodic solution.Furthermore, the positive equilibrium point E * is globally stable in R 4 + .

Discussion
In natural environment, plants are attacked simultaneously by herbivorous insects and pathogen.
Though recent research work mainly focused on either plant-pathogen or plant-insect interactions.In respond to multiple attack, plant emit VOCs that attract natural enemies of herbivores which is important in planning crop plants with better protection against herbivores.Recently, it is observed that rapid usage of pesticides and fertilizers cause major damage on farmland.
So it is important to reduce the requirement for harmful pesticides for insect control.The role of VOCs in mediating tritrophic interactions under multiple attack situation is still unexplored.
In this work, we have tried to investigate the dynamics of plant-herbivore system in the presence of pathogen through mathematical model.The model includes carnivores as an indirect defence against herbivores.In our model, we assumed that herbivore preferentially feeds on the susceptible plant.This induces that infected plant remains in the system due to less attack by herbivore.Many herbivorous insects avoid infected plant [18] and such preferences may allow infected plants to survive until killed by disease.We note that disease is more likely to persist when the herbivores consume susceptible plant only.Our model is described by the four ordinary differential equations.Then we studied the dynamical behavior of the system at various equilibrium points and the stability of those equilibrium points.We obtain five equilibrium points of which E 0 is the population free equilibrium point.The system cannot collapse for any parametric values as the population free equilibrium point is never stable.The planar equilibrium point E 14 always exist which may be stable or unstable under certain restrictions on the system parameters.When the disease transmission rate exceeds a certain threshold value, herbivore free equilibrium point E 124 appears and it becomes stable as long as herbivore consumption rate remains below a certain threshold value.The disease free equilibrium point E 134 can be stable for low disease transmission rate.Coexistence has shown to be possible in the case disease affecting the plant population.Global stability aspect of the positive equilibrium is developed.From biological point of view, this study gives the condition, written in terms of the parameters of the system, under which the disease cannot be eliminated from the community.
Here, we employed the geometric approach developed by Li and Muldowney to global stability for four dimensional system.

Conflict of Interests
The author declares that there is no conflict of interests.