SPATIAL COMPETITION MATHEMATICAL MODEL ANALYSIS FOR THE INVASION , REMOVAL OF KAPPAPHYCUS ALGAE IN GULF OF MANNAR WITH PROPAGATION DELAYS

The aim of this paper is to study the impact of manual removal of Kappaphycus Alvarezii (KA) from the corals in the region of Gulf of Mannar (GoM) and the spatial competition between KA and Native Algae (NA) on coral with propagation delays of KA and NA. A non-linear system of ordinary differential equation (ODE) model has been developed to study the linear stability and the impact of delay terms in the stability at an equilibrium point.


Introduction
The community structure of sedentary organisms is largely controlled by the outcome of direct competition for space.Reefs are one such areas of intense competition between sessile benthic organisms [25].Sufficient access to space and light is crucial for survival on the reef.
The ability to establish, maintain and extend territory (i.e., to outcompete fellow benthic organisms) can affect the composition, size, and distribution of organisms on the benthos [11].In reefs, competition between corals and benthic algae is important in determining the fundamental structure and during coral-algae phase shifts [13].The appearance of corals on substrate previously occupied by corals is often interpreted as evidence that algae actively compete with corals for space.Along with native algae, the exotic algae such as KA also compete for space.
The faster growth rate makes it able to easily overgrow and outcompete NA and coral species.
Understanding the factors defining competitive outcomes among neighbors is thus critical for predicting large-scale changes, such as transitions to alternate states within coral reefs.
Lotka-Volterra systems of differential equations have played a significant role in the development of mathematical ecology.Lotka-Volterra models lack the complexity and realism of more recently developed population models, but their generality makes them a convenient starting point for analyzing ecological systems.Lotka-Volterra systems have assisted as a basis for the development of more realistic models that entangle ratio-dependent functional responses.
Mathematical models have been developed to analyze the three way compettition between sponges, macroalgae and coral to determine the reproduction of benthic competitors [23], parrotfish exploitation [3], sufficiently small levels of fishing mortality [4], grazing intensity [2] and the role of sponge competition on coral reef alternative steady states [24].
We reported the shifting of algal dominated reef ecosystem due to the invasion of KA in Gulf of Mannar [5].Subsequently, the dominance of KA over NA and corals in competing for space has also been reported.To simulate the three way competition among corals, KA and NA, we propose a model as a system of non-linear ODE's.Let x, y and z represent the percent cover of coral, KA and NA respectively and it is assumed that x+y+z = 1, where x -Coral growth with respect to time (t), y -KA growth with respect to time (t), z -NA growth with respect to time (t), 1-(x+y+z) -Turf growth with respect to time (t), r -coral growth on turf (cm 2 /cm 2 /15d), a 1 -KA growth on coral (cm 2 /cm 2 /15d), a 2 -NA growth on coral (cm 2 /cm 2 /15d), a 3 -KA growth on NA (cm 2 /cm 2 /15d), ν -KA growth on turf (cm 2 /cm 2 /15d), h -NA growth on turf (cm 2 /cm 2 /15d).
The manual removal of KA from the coral reefs has been reported [5], we have incorporated the manual removal rate term d of KA into the above model.
where d is the manual removal rate of KA.KA sexual reproduction by spores in the Gulf of Mannar Marine Biosphere Reserve (GoM) in future, when environmental conditions unanimously favor this alga has been deliberated [6].
NA can produce sexually and asexually by forming flagellate and sometimes non-flagellate spores.The vegetative propagation is achieved through fragmentation has been reported [12].
To analyze the impact of delays in propagation of KA and NA through vegetation/spores on coral-algae interactions, we propose the following delay model: where τ 1 -time delay of propagation of KA through vegetation/spores, τ 2 -time delay of propagation of NA through vegetation/spores.
Stability analysis of the model for both non-delay and delay systems is investigated by Sandip Banarjee et.al., [9].Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solution bifurcating from Hopf bifurcation are obtained [10].The local stability of models involving delay dependent parameters has been detailed [15].Periodic oscillations in leukopoiesis models with two delays has been reported [17].Stability analysis of delay differential equation (DDE) model with two discrete delays has been investigated [19,20].Analysis of the bifurcation due to the introduction of the delay term can be reduced to finding whether a related polynomial equation has positive real roots has been discussed [21].
A decomposition technique to investigate the stability of some exponential polynomials, that is, to find conditions under which all zeros of the exponential polynomials have negative real parts has been reported [22].Stability analysis of a multiteam prey-predator model and twoprey one-predator model has been discussed [1,16].A delay system is absolutely stable if it is asymptotically stable for all values of the delays has been reported [27].Several studies on stability analysis of HIV model with delay terms have been documented by many authors [7,8,14,18,26].
In section 2, we examined about equilibria of the system and the existence of the equilibria.Section 3 shows that the existence of stability analysis at different equilibria and also we have proved the existence and uniqueness theorem for the equilibria which capture stability.In section 4, stability analysis of the system of Eqs.( 7)- (9) have been discussed at E 3 (0, y * , 0).In section 5, numerical simulations have been executed.The paper ends with a conclusion.

Equilibria of the system
In this section we discussed about the existence of equilibria of the system of Eqs.( 4)- (6).
Proof.The Jacobian matrix of the system of Eqs. ( 4)-( 6) is given by, The required characteristic equation at E 0 (0,0,0) is given by The roots of the characteristic equation is r, υ − d and h which has two positive eigen values.So, the system is unstable around the trivial equilibrium E 0 (0,0,0).
Proof.The roots of the characteristic equation of the system of Eqs. ( 4)-( 6) at E 2 (1,0,0) are -r, a 1 − d and a 2 which has one positive eigen value.So, the system is unstable around the equilibrium E 2 (1,0,0).
Theorem 3.3.(Existence and Uniqueness) The system of Eqs.( 4)-( 6) is stable around the equilibria E 1 , E 4 and E 5 and it has a unique solution under the conditions given in the proof.
Proof.The proof of the theorem is from the following lemma.
We now focus on the stability of the system of Eqs.( 7)-( 9).
The Jacobian matrix is given by, The required characteristic equation at the equilibrium E 3 (0, y * ,0)is, which gives, In this case the characteristic equation ( 10) reduces to, H) > 0, then the system (7-9) without delay will be locally asymptotically stable around E 3 (0, y * , 0).
Separating the real and imaginary parts we get, Squaring and adding ( 14) and (15), 0 hold then Eq.( 16) has no positive roots.

Conclusion
We have analyzed the effect of two time delays on the coral reef system with the manual removal rate of KA.The actual manual removal rate of KA is very much lesser than our vision in the field.For non-delay case the growth rate of coral increases after the manual removal rate of KA.Then, we have considered the delay case.In the presence of the time delay of propagation of KA through vegetation/spores, the growth rate of KA increases.In the absence of the time delay of propagation of KA through vegetation/spores, the time evolution of three populations are asymptotiocally stable.In the presence of both the time delays (τ 1 , τ 2 ), the system is asymptotically stable.Finally, the impact of manual removal rate of KA triggers its growth.

Fig 1 shows
that the impact of the coral, KA and NA without delays after the manual removal rate of KA.Fig 2 shows that growth rate of KA when τ 1 = 4 and there is no time delay of propagation of NA through vegetation/spores.