1.
INTRODUCTION:
Biomathematics
is an interdisciplinary subject with a vast and exponentially growing
literature pertaining to different disciplines. A large number of mathematical
models have been developed to get an insight into complex biological,
ecological and physiological situations. A variety of Mathematical techniques
have been employed to solve these models. These include techniques for the
solution of differential, difference, integral delay differential and
integro-differential equations as well as useful techniques of linear,
non-linear, dynamic, stochastic programming, calculus of variations, maximum
principle and so on. Some of the latest results of algebraic topology, fuzzy
sets and catastrophe theory have been successfully employed to probe deeply
into the problems of life sciences.
Quite
often, the term Biomathematics is used for Mathematical Biosciences. However,
this term is sometimes is used in a narrow sense of mathematics for biologists
and medical scientists; that is, for a collection of mathematical techniques
especially applicable for biosciences. However the terms Biomathematics and
Mathematical Biosciences are often used synonymously. The terms Mathematical
Biology, Theoretical Biology or mathematical life sciences are also used, but
the term mathematical bioscience is most appropriate and has wider scope. It
includes Mathematical Ecology, Mathematical Demography, Mathematical
Bio-economics, Mathematical medical sciences and Mathematical agriculture.
Mathematical
Biosciences mainly deals with mathematical modelling in Biology and Medicine
and those areas of Biosciences which have already been mathematicized, that is,
those areas which are successful mathematical models are available. To gain a
better insight and help us to deepen an understanding of those areas which have
already been mathematicized, mathematical biosciences constantly endeavours to
widen those areas to which mathematical techniques can be applied with success.
There appears to be a need for widening and deepening the scope of mathematical
biosciences , by utilizing not only the mathematical techniques that are
already used and are known but also to evolve new mathematical methods for
dealing with the complex situations in life sciences.
It
is well known that, physical, economical, engineering and biological problems
can be reduced to the task of solving initial value problems for ordinary
differential equations. We need ways to construct the requisite theories and
formulations, approximations and reductions to the solutions of such problems.
Particularly, problems of Mathematical Biology and Mathematical Physics give us
equations which lead to partial differential equations, integro-differential
equations, differential-difference equations and equations of more complex type
also. Even the simplest equations are generally non- linear in nature. The
technique of quasi linearization enables us to find the form of solutions
through linear approximations. The approximations are carefully constructed to
yield rapid convergence and monotonically as well.
One
of the problems of Mathematical Biosciences is the stability or steadiness of
the system. It is clear that only stable systems may exist for a long time. The
stability depends on whether differential equations of the model are linear or
non- linear. These equations specify the growth rate of each species population
(not per capita growth rate but that of the whole population) as a function of
the sizes of the various interacting populations. Stability also depends on
whether the equations are assumed to apply overall conceivable combinations of
population sizes or only in the neighbourhood of an equilibrium point at which
all growth rates are simultaneously zero. In the former case stability is
called global stability and in the latter case it is called local stability.
If
the differential equations are linear, we may test for global stability by
means of Routh-Hurwitz criteria. If the differential equations are non-linear,
we may treat them as approximately linear in a sufficiently small neighbourhood
of the equilibrium point and can then use Routh-Hurwitz criteria to judge
their local stability in that neighbourhood. While studying the qualitative
stability criteria we exploit Lyapunov’s direct method. A suitable positive
definite function 
is
defined and in a neighbourhood of the equilibrium point, calculate 
(derivative
of).
If is
negative definite it is known that the system is asymptotically stable. These
facts have been utilized to discuss stability criteria of certain biological
systems in this work.
2.
Specific Solutions of Lotka-Volterra Equations:
It
is well known that due to the deterministic nature of the processes involved in
the zones of Biological and Physical sciences, a mathematical description of
these processes generally gives rise to non-linear differential equations. In
recent years much attention has been given to determine explicit solutions of
prey-predator (host-parasite) systems and the existence of positive periodic
solutions of differential equations in non-linear systems, studied by many
authors ([1]- [6]).
7.
CONCLUSIONS:
We
reviewed various research articles in the area of prey predator interactions in
ecosystem in a systematic way. The authors considered various mathematical
models and investigated the stability about the equilibrium points. They also
identified the optimal harvesting strategies which are most useful for
fishermen. Some of authors also computed the population intensities of
fluctuations (variances) around the positive equilibrium due to white noise.
8.
FUTURE WORKS:
Following the same symbolization, we can incorporate
diffusive terms in almost all the models which are quoted in the above to study
the dynamics of the systems with respect to time and space variables. The
effect of dispersal and spatial heterogeneity is more important in the
population dynamics, as it plays major role on the stability of the ecosystems.
The diffusive analysis may produce interesting results that the effect of
diffusion coefficients in changing the unstable behaviour to stable one. It is
also very important to analyse the dynamical features of the various models and
to get an insight on the switch of stability in the presence of cross
diffusion. We can also improve the model (4) to find the steady states by using
various algorithms and numerical methods.
9.
REFERENCES:
1. A.Y. Wang and D.Q. Jiang, Existence of positive
periodic solutions for functional differential equations, Kyush J Math. 2002;
56: 193-202.
2. B.W. Liu and L.H. Huang. Existence and uniqueness of
periodic solutions for a kind of first order neutral functional differential
equations. J Math Anal Appl. 2006; 322: 121-132.
3. D.Q. Jiang and J. J. Wei, Existence of positive
periodic solutions for Volterra intergo-differential equations, Acta Math
Sincia B. 2001; 21: 553-560.
4. L.S. Luckinbill, Coexistence in laboratory populations
of paramecium aurelia and its predator didinium nasutum, Journal of Ecology.
1973; 54 : 1320-1327.
5. M.A. Krasnoselskii, Positive Solutions of Operator
Equations, Gorninggen: Noordhoff, 1964.
6. S.P. Lu, On the existence of positive periodic
solutions for neutral functional differential equation with multiple deviating
arguments. J. Math Anal Appl. 2003; 280: 321—333.
7. V.S. Varma, Exact solutions for a special Prey Predator
competing species system, Bull. Math. Biol. 1977; 39: 619-622.
8. A.J. Willson, On Varma’s Prey-Predator problem, Bull.
Math. Biol. 1980; 42: 599- 600.
9. R.R. Burnside, A note on exact solutions of
Prey-Predator equations, Bull. Math. Biol. 1982; 44: 893-897.
10. K.N. Murty and D.V.G. Rao, Approximate analytical
solutions of general Lotka-Volterra equations, J. Math. Anal. Applications.
1987; 122: 582-588.
11. K.N. Murty, K.R. Prasad and M.A.S. Srinivas, Certain
mathematical models for biological systems and their approximate analytical
solutions, Proceedings of the international Symposium on Non-linear analysis
and application to Biomathematics, Andhra University, Visakhapatnam. 1987;
146-155.
12. W.T. Schoener, Alternatives to Lotka-Volterra
competition Models of Intermediate complexity, Theoret. Population Biol. 1976;
10: 309-333.
13. Younhee Ko, The asymptotic stability behavior in a
Lotka-Volterra type predator-prey system, Bull. Korean Math. Soc. 2006; 43:
575-587.
14. F.J. Ayala, M.E. Gilpin and J.G. Ehrenfeld, Competition
between species: Theoretical models and experimental tests, Theoret. Population
Biol. 1973; 4: 331-356.
15. M.E Gilpin and F.J.Ayala, Global models of growth and
competition, Acad.Sci. 1974; 3590-3593.
16. B. Dubey, P.Chandra and P. Sinha, A model for fishery
resource with reserve area, Non-linear analysis: Real world applications.
2003; 4: 625-637.
17. Tapan Kumar Kar and Swarnakamal Misra, Influence of
prey reserve in a prey-predator fishery, Non-Linear Analysis. 2006; 65:
1725-1735.
18. Wendi Wang, Yasuhiro Takeeuchi, Yasuhisa Saito and
Shinji Nakaoka, Prey-predator system with parental care for predators, Journal
of Theoretical Biology. 2006; 241: 451-458.
19. Rui Zhang, Junfang Sun and Haixia Yang Analysis of a
prey-predator fishery model with prey reserve, Appl.Math.Sciences. 2007; 1:
2481-2492.
20. M.A.S. Srinivas, Y. Narasimhulu, M.N. Srinivas, “A Prey
– Predator Model for Fishery Resource, International Journal of Mathematical
Sciences and Engg. Applications. 2008; 2(III) : 173-189.
21. M.N. Srinivas, M.A.S. Srinivas, Kalyan Das, Nurul Huda
Gazi, Prey– Predator Fishery Model with Stage Structure in Two Patchy Marine
Aquatic Environment, Applied Mathematics (Scientific Research) (USA). 2011; 2:
1405-1416.
22. Kalyan Das, M.N. Srinivas, M.A.S. Srinivas, N.H.Gazi,
Chaotic dynamics of a three species prey-predator competition model with
bionomic harvesting due to delayed environmental Noise as external driving
force, C R Biologies. 2012; 335: 503-513.
23. Kalyan Das, K. Shiva Reddy, M. N. Srinivas, N.H. Gazi,
Chaotic dynamics of a three species prey–predator competition model with noise
in ecology, Applied Mathematics and Computation. 2014; 231: 117-133.
