7. Discussion
In predator-prey interaction, predation is considered to be the main
force that promotes coexistence of competing species by reducing the
strength of competition [6]. If the predator chooses strongest
competitor species, mostly then it relives competition pressure on other
species, thereby allowing coexistence of multiple species. Recent field
experiments showed that predators can induce a non-consumptive effect on
their prey, for example fear [23]. Due to predation fear, prey can
adopt defensive strategies that disrupt coexistence [20].To address
fears induced coexistence on competing species, we developed a
mathematical model of two competing prey species and one predator where
predator, not only kill both the prey but also shows non-consumptive
effect upon them. Our system also includes intraspecific competition
within the predator population. Takeuchi and Adachi [24] addresses
an ecological system with the same type of species, but no fear effect,
nor intraspecific competition within the predator populations obtaining
coexistence results. The proposed model is biologically meaningful in
the sense that any positive solution initiating in the positive orthant
remains both non-negative and bounded.
Mathematical analysis of the model established that the system cannot
collapse for any parameter value as the origin is always unstable. If
the second prey has low intrinsic growth rate and the predator has a
high death rate then the predator cannot prevent the first prey and
tends to its carrying capacity;\(E_{1}\) is an attractor whereas
the opposite hold if the first prey has low intrinsic growth rate. If
the intraspecific competition if stronger than the interspecific
competition and the predator has a high death rate then both the prey
can coexist at \(E_{12}\) while predator population goes to extinction
due to large death rate. The first prey and the predator can coexist at\(E_{13}\) when the second prey has moderate intrinsic growth rate.
Again the second prey and the predator can coexist at \(E_{23}\ \)as
long as the intrinsic growth rate remains below a certain threshold
value. Using invasion analysis, we derived criterion for uniform
persistence of our model system that ensures the existence of positive
(coexistence) equilibrium point. Local stability of the coexistence
equilibrium point is possible if the ratio of intake capacity by the
predator lie within an interval. The existence of Hopf bifurcation is
shown by considering the level of fear as bifurcation parameter. The
nature of limit cycle emerging through a Hopf bifurcation is predicted
by calculating the coefficient of curvature of the limit cycle. If the
intraspecific competition of the first prey is less than that of second
prey then supercritical limit cycle appears. In this paper we have not
considered intraspecific competitive rate \(h\) as a bifurcation
parameter. But one obtains the bifurcation result for taking \(h\) as
bifurcation parameter. When most of the predators are involved in
intraspecific competition, stable coexistence increases (see Fig. 1e).
The novelty of our work is the inclusion of fear effect and
intraspecific competition within the predator populations which are not
considered in [24]. This investigation generalizes the existing
knowledge of fear effect of predator on single prey species [16-18,
27, 28, 30]. As high level of fear can destroy coexistence that agrees
with [20]still coexistence of predator and competing prey is
possible with the increase of intraspecific competition within the
predator population. Our theoretical observations will be helpful to
verify some experimental data set of two competing prey and one predator
system.
It may also be worthwhile to see how the other response function rather
than Holling type I affects the dynamics of the system. From
experimental observation, we have considered the fear effect on
reproduction term of prey population still it is reasonable to see the
fear effect on intraspecific, interspecific competition or death rate of
prey populations.
References .
[1] P. Chesson, J. J. Kuang, The interaction between predation and
competition. Nature. 456 (2008) 235-238.
[2] S. Creel, D. Christianson, Relationships between direct
predation and risk effects. Trends Ecol. Evol. 23 (2008) 194-201.
[3] S. Creel, D. Christianson,
S. Liley, J. A. Winne, predation risk affects reproductive physiology
and demography of elk, Science 315 (2007) 960-960.
[4] W. Cresswell, Non-lethal effects of predation risk in birds.
Ibis. 150 (2008) 3-17.
[5] W. Cresswell, Predation in bird populations, Journal of
Orinthology 152 (2011) 251-263.
[6] J. Gurevitch, J. A. Morrison, L. V. Hedges, The interaction
between competition and predation: a meta-analysis of field experiments.
American Naturalist. 155 (2000) 435-453.
[7] R. D. Holt, Predation, apparent competition, and structure of
prey communities. Theor. Pop. Biol. 12 (1977) 197-229.
[8] R. D. Holt, Spatial heterogeneity, indirect interactions, and
the coexistence of prey species. American Naturalist. 124 (1984)
377-406.
[9] V. Hutson, A theorem on average Lyapunov function, Monatsh Math.
98 (1984) 267-275.
[10] V. Hutson, The existence of an equilibrium for permanent
systems, Rocky Mountain journal of Mathematics. 20(1990) 1033-1040.
[11] B. P. Kotler, R. D. Holt, Predation and competition: the
interaction of two types of species interactions. Oikos. 54 (1989)
256-260.
[12] V. Krivan, Competitive co-existence caused by adaptive
predators. Evolutionary Ecology Research. 5 (2003) 1163-1182.
[13]J. J. Kuang, P. Chesson, Interacting coexistence mechanisms in
annual plant communities: frequency-dependent predation and the storage
effect. Theor. Pop. Biol. 77 (2010) 56-70.
[14] S. L. Lima, Nonlethal effects in the ecology of predator-prey
interactions-what are the ecological effects of anti-predator decision
making? Bioscience. 48 (1998) 25-34.
[15] W. M. Liu, Criterion of Hopf bifurcation without using
eigenvalues, J. Math. Anal. Appl. 182 (1994) 250-256.
[16] S. Mondal, A. Maiti, G. P. Samanta, Effects of fear and
additional food in a delayed predator-prey model, Biophysical Reviews
and Letters. 13 (2018) 157-177.
[17] D. Mukherjee, Study of fear mechanism in predator-prey system
in the presence of competitor for the prey. Ecol. Genetics and Genomics.
15(2020) 10052.
[18] S. Pal, S. Majhi, S. Mandal, N. Pal, Role of fear in a
predator-prey model with Beddington-DeAngelis functional response, Z.
Naturforsch. 74 (2019) 581-585.
[19] P. Panday, N. Pal, S. Samanta, J. Chattopadhyay, Stability and
bifurcation analysis of a three-species food chain model with fear, Int.
J. Bifurcation Chaos. 28 (2008).
[20] R. M. Pringle et al., Predator-induced collapse of niche
structure and species coexistence. Nature. 570 (2019) 58-64.
[21] Z. Qiu, Dynamics of a model for virulent phase T4, J. Biol.
Syst. 16 (2008) 597-611.
[22] S. Sasmal, Y. Takeuchi, Dynamics of a predator-prey system with
fear and group defense. J. Math. Anal Appl. (2019). Doi:
org/10.1016/j.jmaa.2019, 123471. Model. 64 (2018) 1-14.
[23] O. J. Schmitz, Predators affect competitors’ coexistence
through fear effects. Nature.570 (2019) 43-44.
[24] Y. Takeuchi, N. Adachi, Existence and bifurcation of stable
equilibrium in two-prey, one-predator communities. Bull. Math. Biol. 6
(1983) 877-900.
[25] R. R. Vance, Predation and resource partitioning in one
predator-two prey model communities, American Naturalist. 112 (1978)
797-813.
[26] X. Wang, L. Deng, W. Zhang, Hopf bifurcation analysis and
amplitude control of the modified Lorenz system. Appl. Math. Comp. 225
(2013) 333-344.
[27] X. Wang, L. Zanette, X.
Zou, Modelling the fear effect in predator-prey interactions, J. Math.
Biol. 73 (2016) 1179-1204.
[28] X. Wang, X. Zou, Modelling the fear effect in predator-prey
interactions with adaptive avoidance of predators. Bull. Math. Biol. 79
(2017) 1-35.
[29] L. Y. Zanette, A. F. White, M. C. Allen, C. Michael, Perceived
predation risk reduces the number of offspring songbirds produce per
year. Science. 334 (2011) 1398-1401
[30] H. Zhang, Y. Cai, S. Fu, W. Wang, Impact of the fear effect in
a prey-predator model incorporating a prey refuge, Appl. Math. Comp. 356
(2019) 328-337.