  Forest-fire and avalanche models support the notion that frequent
catastrophes prevent the growth of very large populations and as such prevent
rare large-scale catastrophes. We show that this notion is not universal. A new
model class leads to a paradigm shift in the influence of catastrophes on the
family-size distribution of sub-populations. We study a simple population
dynamics model where individuals, as well as a whole family, may die with a
constant probability, accompanied by a logistic population growth model. We
compute the characteristics of the family-size distribution in steady-state and
the phase diagram of the steady-state distribution, and show that the family
and catastrophe size variances increase with the catastrophe frequency, which
is the opposite of common intuition. Frequent catastrophes are balanced by a
larger net-growth rate in surviving families, leading to the exponential growth
of these families. When the catastrophe rate is further increased, a second
phase transition to extinction occurs, when the rate of new families creations
is lower than their destruction rate by catastrophes.
