Additional food causes predator "explosion" -- unless the predators compete

The literature posits that an introduced predator population, is able to drive it's target pest population extinct, if supplemented with high quality additional food of quantity $\xi > \xi_{critical}$, \cite{SP11, SPV18, SPD17, SPM13}. We show this approach leads to infinite time blow-up of the predator population. We propose an alternate model in which the additional food induces predator competition. Analysis herein indicates that there are threshold values $c^{*}_{1} < c^{*}_{2} < c^{*}_{3}$ of the competition parameter $c$, s.t. when $c < c^{*}_{1}$, the pest free state is globally stable, when $c^{*}_{2} < c < c^{*}_{3}$, bi-stability is possible, and when $c^{*}_{3} < c$, up to three interior equilibriums could exist. As $c$ and $\xi$-$c$ are varied, standard co-dimension one and co-dimension two bifurcations are observed. The recent dynamical systems literature involving predator competition, report several non-standard bifurcations such as the saddle-node-transcritical bifurcation (SNTC) occurring in co-dimension two \cite{KSV10, BS07}, and cusp-transcritical bifurcation (CPTC) in co-dimension three, \cite{D20, BS07}. We show that in our model structural symmetries can be exploited to construct a SNTC in co-dimension two, and a CPTC also in co-dimension two. We further use these symmetries to construct a novel pitchfork-transcritical bifurcation (PTC) in co-dimension two, thus explicitly characterizing a new organizing center of the model. Dynamics such as homoclinic orbits, concurrently occurring limit cycles, and competition driven Turing patterns are also observed. Our findings indicate that increasing additional food in predator-pest models, can hinder bio-control, contrary to some of the literature. However, additional food that also induces predator competition, leads to novel bio-control scenarios, and complements the work in \cite{H21, B98, K04, D20, BS07, VH19}.