Vitalie Ciubotaru

Previous posts on this topic:

• A[nother] strange idea: browser game
• The Game: progress (1)
• The Game: progress (2)
• The Game: progress (3)
• The Game: progress (4)
• The Game: progress (5)
• The Game: universality vs. specialization of players
• The Game: life and death
• The Game: plant types and species

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1. Plant size

Under normal conditions, plant size increases over time. Plant growth depends on:

• Plant’s current size () — the larger the plant is, the more it adds every day.
• Growth rate () — a general concept, which will be discussed in more detail below.
• Cell size () — i.e. there is an upper limit for growth.

Thus, . When the plant is small, the upper limit is not important and the limiting factor is current size, so initially the plant grows [almost] exponentially. When the plant grows large, the upper limit becomes more binding and growth decelerates again. Let me solve this equation for .

Solving the left side integral:

So
Solving the right side integral:

Now putting it all together:

Simplify and redefine the constant:



Finally, making it discrete:

2. Growth rate

Time to discuss . The growth rate depends on:

• (mis)match between the “climate requirements” of a particular species and “climate” or “resource endowment” of a particular cell
• capacity to acquire resources — roots and leaves

For example, insufficient sunlight can be partly compensated by large leaves and insufficient minerals/water — by longer roots. Excessive resources will represent a small saving in roots/leaves, but will still be limiting the growth rate. As a possible way to make the impact of deficit and excess of resources to be symmetric, developing roots and leaves can cost no resources, just a mouse click and some waiting time. Have to think a bit on this.

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