  Aphids are damaging insect pests on many crops. Their density can rapidly
build up on a host plant to several thousand over one growing season.
Occasionally, a competition-driven decline in population early in the season,
followed by a build-up later, is observed in the field. Such dynamics cannot be
captured via standard models, such as introduced in Kindlmann and Dixon, 2010.
In Kindlmann et al., 2007, a logistic non-local population model with variable
carrying capacity is proposed to capture these alternative dynamics. The
proposed model has a rich dynamical structure and can predict multiple
population peaks, as observed in the field. We show that additionally, this
model also possesses solutions that can blow-up/explode in finite time. The
blow-up is seen to occur for both large and small initial conditions and can
occur both early and late in the season. We propose a model extension that,
under certain parametric restrictions, has global time-bounded solutions. We
discuss the ecological applications of our findings.
