The Model. Invasion Threshold

The two-dimensional continuous-time mesoscopic equations for each stage are 

Here P x, t), 5i(x, t), and 2(x, t) are the number density of plants, unripe seeds, and mature seeds located at point x at time t, respectively, while p(x, t), Si(x, t), and j 2(x, 0 are the number density of plants, unripe and dispersible seeds arriving at point X at time t, respectively. The distribution functions y3 (t), (t), and cp [t) 

For the steady state to be biologically meaningful, the population density must be positive, which requires 

Here fix, t) incorporates all the terms where initial conditions appear, and Pit) can be regarded as the PDF of times between successive generations. It is defined as 

The roles of the PDFs p, P2, P3, and are encapsulated in Pit). Given a PDF for the generation of seeds and plant mortality we can calculate from (7.57) the minimum number of seeds that need to be produced by a plant to invade successfully. This outcome can sometimes be an alternative to the well-known model by Caswell [72] based on population matrices for predicting the population growth characteristics. The product Yuq is approximately equivalent to the parameter k of Caswell s model, which usually represents the population growth in the matrix formalism. Compared with Caswell s approach, where the condition k 1 determines the invasive character of a species, we find that the threshold depends on the distributions of plant survival p it) and seed production P2U). Therefore, this model can be seen as a generalization of Caswell s result for the case where temporal PDFs for every 

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