Competition–colonization trade-off

${\displaystyle {\frac {{\text{d}}p_{1}}{{\text{d}}t}}=c_{1}p_{1}(1-p_{1})-m_{1}p_{1}}$

${\displaystyle {\frac {{\text{d}}p_{2}}{{\text{d}}t}}=c_{2}p_{2}(1-p_{1}-p_{2})-m_{2}p_{2}-c_{1}p_{1}p_{2}}$

Where: ${\displaystyle p_{i}=}$ fraction of patches that are occupied by species ${\displaystyle i}$;

${\displaystyle c_{i}=}$ colonization rate of species ${\displaystyle i}$;

${\displaystyle m_{i}=}$ mortality rate of species ${\displaystyle i}$ (independent of patch density).

Species 1 = competitor, can colonize in area that is uninhabited or inhabited by species 2 ${\displaystyle (1-p_{1})}$.

Species 2 = colonizer, can only colonize in uninhabited areas ${\displaystyle (1-p_{1}-p_{2})}$.

Species 2 is subject to displacement by its competitor ${\displaystyle (-c_{1}p_{1}p_{2})}$.

If species 2 has a higher colonization rate it can coexist with species 1: ${\displaystyle c_{2}>{\frac {c_{1}(c_{1}+m_{2}-m_{1})}{m_{1}}}}$.

This model is described as the displacement competition model, it has been observed in marine mollusks and fungi. This model makes two large assumptions: 1. "a propagule of a superior competitor takes over a patch from an adult of the inferior competitor".[6] 2. The adult must be displaced fast enough to ensure that it does not reproduce while it is being displaced.

${\displaystyle {\frac {{\text{d}}p_{1}}{{\text{d}}t}}=c_{1}{\frac {f}{f+p_{2}}}p_{1}(h-p_{1}-p_{2})m_{1}p_{1}}$

${\displaystyle {\frac {{\text{d}}p_{2}}{{\text{d}}t}}=c_{2}{\frac {g}{g+p_{1}}}p_{2}(h-p_{1}-p_{2})m_{2}p_{2}}$

Colonization rate is now described by interspecific competition.

${\displaystyle {\frac {f}{f+p_{2}}}}$ and ${\displaystyle {\frac {g}{g+p_{1}}}}$. Both f and g > 0.

An increase in p1 is related to a decrease in the colonization rate of species 2.

g < f implies a competitive advantage of species 1 and c2 > c1 implies a colonization advantage for species 2.