# 2UP

The 2Up model is a spatially explicit global urbanisation and population growth model.

# Indices

• $$i$$: Land Unit, a 30" x 30" lat long part.
• $$c$$: Country
• $$t$$: Time step: {2010, 2020, 2030, 2040, 2050, 2060, 2070, 2080}

# Urban Allocation

Static variables

• $$I^i_c$$: Incidence matrix; for each $$i$$: $$\sum\limits_c I^i_c = 1$$
• $$A_i$$: Area of land unit $$i$$

Dynamic variables

• $$\sideset{^t}{_i}S$$: Suitability, a function or combination of spatially explicit suitability factors
• $$\sideset{^t}{^c}P$$: Projected population, source: SSP database - Institute of Applied Systems Analysis (IIASA) - [1]
• $$\sideset{^t}{^c}D$$: Hyde population density index, source: History Database of the Global Environment - Klein Goldewijk, Dr. ir. C.G.M. (Utrecht University) (2017): Anthropogenic land-use estimates for the Holocene; HYDE 3.2. DANS. [2]
• $$\sideset{^t}{^c}C$$: Urban Claim

Urban Claim

• $$\sideset{^t}{^c}U := \sideset{^{2010}}{^c}U \cdot {\sideset{^t}{^c}P \over \sideset{^{2010}}{^c}P} \cdot {\sideset{^{2010}}{^c}D \over \sideset{^t}{^c}D}$$
• $$\sideset{^t}{^c}C := \min(\sideset{^t}{^c}U, \sum\limits_i{A_i \cdot I^i_c})$$ Claim is U, but no more than the available land

Urban Spatial Allocation

• $$X_i \in \{0, 1\}$$ such that $$\sum\limits S_i \cdot X_i$$ is maximal and that for each $$c$$: $$\sum\limits_i X_i \cdot A_i \cdot I^i_c = C^c$$ where $$X_i = 1$$ represents projected urban land use.

This allocation is done by taking the $$C^c \over \sum\limits_i A_i \cdot I^i_c$$ th percentile of the land units of $$c$$, descendingly ordered by $$S_i$$ and weighted by $$A_i$$.

# Population Projection

Projected population growth $$\sideset{^t}{^c}{EP} := \sideset{^t}{^c}P - \sideset{^{t-1}}{^c}P$$ is spatially allocated according to the same suitability $$\sideset{^t}{_i}S$$, taking into account a spatially explicit maximum population density $$\sideset{^t}{_i}{MD}$$.

• $$\sideset{^t}{_i}{MD} := \max\left(\sqrt{\sum\limits_{j \in W(i)} \left( \sideset{^{t-1}}{_i}P \over A_i \right)^2 \cdot \sideset{^t}{_i}X \cdot {dd}_{ij} \over \sum\limits_{j \in W(i)} \sideset{^t}{_i}X \cdot {dd}_{ij}}, \sum\limits_c { I^i_c \cdot {\sum\limits_i \sideset{^{t-1}}{_i}P \cdot \sideset{^t}{_i}X \cdot I^i_c \over \sum\limits_i A_i \cdot \sideset{^t}{_i}X \cdot I^i_c} } \right)$$
• $${dd}_{ij}$$: distance decay function.
• $$W(i)$$: A window around land unit $$i$$ such that $${dd}_{ij} \ne 0 \implies j \in W(i)$$.

If $$\sideset{^t}{^c}{EP} > 0$$

• $$\sideset{^t}{_i}\Delta := \sum\limits_c I^i_c \cdot \sideset{^t}{^c}{EP} \cdot { \sideset{^t}{_i}S \cdot A_i \cdot \sideset{^t}{_i}X \cdot (\sideset{^{t-1}}{_i}P < \sideset{^{t-1}}{_i}{MD} * A_i ) \over \sum\limits_i I^i_c \cdot \sideset{^t}{_i}S \cdot A_i \cdot \sideset{^t}{_i}X \cdot (\sideset{^{t-1}}{_i}P < \sideset{^{t-1}}{_i}{MD} * A_i ) }$$