# Density dependent models

Density dependent model classes are now implemented. This vignette will get more details shortly. For now, see the example below:

## Example of a simple, stochastic, kernel-resampled model with density dependence

This example assumes that density dependence is modeled as a fixed effect in survival and recruit production models, and assumes there is no density dependence in growth or probability of reproducing models. The survival ($$s(z, N)$$/s_yr), growth ($$G_{yr}(z',z)$$/ g_yr), and number of recruit models ($$r_{s,yr}(z, N)$$/r_s_yr) have year-specific intercepts as well.

The mathematical form for the IPM is below:

1. $$n(z', t+1) = K_{yr}(z', z, N)n(z, t)dz$$

2. $$N = \int_L^Un(z,t)dz$$

3. $$K_{yr}(z', z, N) = P_{yr}(z', z, N) + F_{yr}(z', z, N)$$

Here, $$N$$ represents the total population size. The kernel values fluctuate as a function of $$N$$ at each iteration of the model.

The $$P_{yr}(z', z, N)$$ kernel is comprised of a density independent function for growth (Eq 6-7) and a density dependent function for survival (Eq 5). $$f_G$$ denotes a Gaussian probability density function:

1. $$P(z', z, N) = s(z, N) * G(z', z)$$

2. $$Logit(s(z, N)) = \alpha_s + \alpha_{s,yr} + \beta_s^z * z + \beta_s^{N} * N$$

3. $$G(z', z, \theta) = f_G(z', \mu_{G,yr}(z), \sigma_G)$$

4. $$\mu_{G,yr}(z) = \alpha_G + \alpha_{G,yr} + \beta_G^z * z$$

The $$F_{yr}(z',z, N)$$ kernel is comprised of a density independent function for recruit size (Eq 10) and probability of reproducing (Eq 9), and a density dependent function for number of recruits produced by parents (Eq 11). $$f_{r_d}$$ denotes a Gaussian probability density function:

1. $$F_{yr}(z', z, N) = r_r(z) * r_{s,yr}(z, N) + r_d(z')$$

2. $$Logit(r_r(z)) = \alpha_{r_r} + \beta_{r_r}^z * z$$

3. $$r_d(z') = f_{r_d}(z', \mu_{r_d}, \sigma_{r_d})$$

4. $$Log(r_{s,yr}(z, N)) = \alpha_{r_s} + \alpha_{{r_s},yr} + \beta_{r_s}^z * z + \beta_{r_s}^N * N$$

We’ll simulate a 50 year time series using hypothetical parameter values. The fixed parameter values are created as with a density independent model. The difference is that we now have two more parameters: s_dd, and r_s_dd. These are the coefficients that correspond to $$\beta_s^N$$ and $$\beta_{r_s}^N$$, respectively. The chunk below initializes the data list object, which we name params.

library(ipmr)

data_list = list(
s_int     = 1.03,
s_slope   = 2.2,
s_dd      = -0.7,
g_int     = 8,
g_slope   = 0.92,
sd_g      = 0.9,
r_r_int   = 0.09,
r_r_slope = 0.05,
r_s_int   = 0.1,
r_s_slope = 0.005,
r_s_dd    = -0.03,
mu_rd     = 9,
sd_rd     = 2
)

# Now, simulate some random intercepts for growth, survival, and offspring production

g_r_int   <- rnorm(5, 0, 0.3)
s_r_int   <- rnorm(5, 0, 0.7)
r_s_r_int <- rnorm(5, 0, 0.2)

nms <- paste("r_", 1:5, sep = "")

names(g_r_int) <- paste("g_", nms, sep = "")
names(s_r_int) <- paste("s_", nms, sep = "")
names(r_s_r_int) <- paste("r_s_", nms, sep = "")

params     <- c(data_list, g_r_int, s_r_int, r_s_r_int)

Next, we initialize the model using init_ipm. The difference is that the second argument is now changed to "dd" to denote that this is a density dependent model.

dd_ipm <- init_ipm(sim_gen = "simple",
di_dd = "dd",
det_stoch = "stoch",
kern_param = "kern")

Once we’ve done that, we’re ready to begin specifying the kernel forms. One previously not mentioned aspect of define_pop_state() is that, in addition to defining initial conditions, 2 additional helper variables are generated: n_stateVariable_t and n_stateVariable_t_1. These can be used to reference the population states in vital rate and/or kernel expressions.

These will look very similar to the ones we specified for density-independent models, except that we now include the term s_dd * sum(n_size_t) in the survival expression. sum(n_size_t) is the syntax ipmr uses to denote total population size. Further down, there is an example of how to use subsets of the trait distribution.

dd_ipm <- define_kernel(
proto_ipm        = dd_ipm,
name             = "P_yr",
formula          = s_yr * g_yr,
family           = "CC",
s_yr             = plogis(s_int + s_r_yr + s_slope * size_1 + s_dd * sum(n_size_t)),
g_yr             = dnorm(size_2, g_mu_yr, sd_g),
g_mu_yr          = g_int + g_r_yr + g_slope * size_1,
data_list        = params,
states           = list(c("size")),
uses_par_sets    = TRUE,
par_set_indices = list(yr = 1:5),
evict_cor        = TRUE,
evict_fun        = truncated_distributions("norm", "g_yr")
) 

Other than the inclusion of the density dependent term in the survival expression, this should look quite similar to the density-independent kernel-resampled models from the Introduction vignette. We are now ready to continue defining the $$F_{yr}(z',z,N)$$ kernel.

dd_ipm <- define_kernel(
proto_ipm        = dd_ipm,
name             = "F_yr",
formula          = r_r * r_s_yr * r_d,
family           = "CC",
r_r              = plogis(r_r_int + r_r_slope * size_1),
r_s_yr           = exp(r_s_int + r_s_r_yr + r_s_slope * size_1 + r_s_dd * sum(n_size_t)),
r_d              = dnorm(size_2, mu_rd, sd_rd),
data_list        = params,
states           = list(c("size")),
uses_par_sets    = TRUE,
par_set_indices = list(yr = 1:5),
evict_cor        = TRUE,
evict_fun        = truncated_distributions("norm", "r_d")
) 

Again, we’ve add the f_s_dd * sum(n_size_t) to the expression for f_s_yr, but otherwise, not much is different from how we’ve defined density independent models. The rest of the model definition process is unchanged.

 dd_ipm <-  dd_ipm %>%
define_impl(
make_impl_args_list(
kernel_names = c("P_yr", "F_yr"),
int_rule     = rep("midpoint", 2),
state_start    = rep("size", 2),
state_end      = rep("size", 2)
)
) %>%
define_domains(
size = c(0, 50, 200)
) %>%
define_pop_state(
n_size = runif(200)
) %>%
make_ipm(
iterate = TRUE,
iterations = 50,
kernel_seq = sample(1:5, 50, replace = TRUE)
)

lambda methods are defined for all density-dependent models as well. It is fairly straightforward to plot population sizes for these models by extracting the column sums of the arrays in pop_state.

time_step_lams <- lambda(dd_ipm, type_lambda = "all")
stoch_lam      <- lambda(dd_ipm, type_lambda = "stochastic", burn_in = 0.15)

pop_sizes <- colSums(dd_ipm$pop_state$n_size)

plot(pop_sizes, type = "l")